2D Geometry: Angles, Shapes, and Area

Angle Measurement and Relationships

Angles are everywhere. You can see them in the corners of a book, the hands of a clock, and where two roads meet. In geometry, an angle is formed when two rays meet at one endpoint.

In this lesson, you will learn how to measure angles and how to use angle relationships to find missing angle measures. These relationships help us solve problems even when we do not know every angle right away.

1. What is an angle?

An angle is made of two rays that share a common endpoint. The common endpoint is called the vertex.

Angles are measured in degrees, written with the symbol \,^\circ. For example, an angle measuring 45 degrees is written as \(45^\circ\).

2. Types of angles by size

• Acute angle: less than \(90^\circ\)
• Right angle: exactly \(90^\circ\)
• Obtuse angle: greater than \(90^\circ\) but less than \(180^\circ\)
• Straight angle: exactly \(180^\circ\)

These angle types help us describe what an angle looks like and check whether an answer makes sense.

3. Measuring angles with a protractor

A protractor is a tool used to measure angles. It is usually shaped like a half-circle and marked with degrees from \(0^\circ\) to \(180^\circ\).

To measure an angle:

1. Place the center point of the protractor on the angle's vertex.
2. Line up one ray with the \(0^\circ\) line on the protractor.
3. Look at where the other ray crosses the numbered scale.
4. Read the correct scale carefully.

Many protractors have two sets of numbers. Start from the side where the ray lines up with \(0^\circ\). Then follow that scale to the other ray.

4. Important angle relationships

Some angles are connected in special ways. If you know one angle, you can often find another.

• Adjacent angles: two angles that share a side and a vertex and are next to each other
• Vertical angles: opposite angles formed by two intersecting lines
• Complementary angles: two angles whose measures add to \(90^\circ\)
• Supplementary angles: two angles whose measures add to \(180^\circ\)

Adjacent angles are side by side. They do not overlap.

Vertical angles are always equal. If one vertical angle measures \(70^\circ\), the angle across from it also measures \(70^\circ\).

Complementary angles make a right angle together. So their total is always:

$$a+b=90$$

Supplementary angles make a straight angle together. So their total is always:

$$a+b=180$$

5. Using equations to find unknown angles

Sometimes an unknown angle is shown with a letter, like \(x\). We can use what we know about angle relationships to write an equation and solve it.

For example, if two angles are complementary and one angle is \(32^\circ\), then:

$$x+32=90$$

Subtract 32 from both sides:

$$x=58$$

So the missing angle is \(58^\circ\).

6. Worked Examples

Example 1: Measuring and naming an angle

A protractor shows that an angle measures \(40^\circ\).

What type of angle is it?

Since \(40^\circ\) is less than \(90^\circ\), it is an acute angle.

Answer: \(40^\circ\), acute angle

Example 2: Complementary angles

Two angles are complementary. One angle measures \(27^\circ\). Find the other angle.

Complementary angles add to \(90^\circ\), so write:

$$x+27=90$$

Now solve:

$$x=90-27$$ $$x=63$$

Answer: The missing angle is \(63^\circ\).

Example 3: Supplementary angles

Two angles form a straight line. One angle measures \(115^\circ\). Find the other angle.

Angles on a straight line are supplementary, so they add to \(180^\circ\):

$$x+115=180$$

Solve:

$$x=180-115$$ $$x=65$$

Answer: The missing angle is \(65^\circ\).

Example 4: Vertical angles with a variable

Two lines cross. One angle is labeled \((3x+10)^\circ\). The vertical angle across from it is labeled \(70^\circ\). Find \(x\).

Vertical angles are equal, so:

$$3x+10=70$$

Subtract 10 from both sides:

$$3x=60$$

Divide both sides by 3:

$$x=20$$

Now check the angle measure:

$$3(20)+10=60+10=70$$

Answer: \(x=20\)

7. Tips for solving angle problems

• First decide what kind of angle relationship you see.
• If angles are vertical, set them equal.
• If angles are complementary, add them to get \(90^\circ\).
• If angles are supplementary, add them to get \(180^\circ\).
• Check whether your answer is reasonable. For example, an acute angle should be less than \(90^\circ\).

8. Common mistakes to avoid

• Mixing up complementary and supplementary angles
• Reading the wrong scale on a protractor
• Forgetting that vertical angles are equal
• Forgetting to include the degree symbol in angle measures

9. Quick review

• Angles are measured in degrees.
• Acute angles are less than \(90^\circ\).
• Right angles are exactly \(90^\circ\).
• Obtuse angles are between \(90^\circ\) and \(180^\circ\).
• Straight angles are exactly \(180^\circ\).
• Vertical angles are equal.
• Complementary angles add to \(90^\circ\).
• Supplementary angles add to \(180^\circ\).

Summary

Angle measurement helps us describe how wide an angle opens. By using a protractor and understanding angle relationships, you can find missing angle measures.

Remember these key facts: vertical angles are equal, complementary angles add to \(90^\circ\), and supplementary angles add to \(180^\circ\). These rules make solving angle problems much easier.

Lesson: Classifying Triangles and the Triangle Inequality Theorem

Triangles are one of the most important shapes in geometry. A triangle has 3 sides, 3 angles, and 3 vertices (corner points).

In this lesson, you will learn two big ideas:

• how to classify triangles by their sides and by their angles
• how to use the Triangle Inequality Theorem to decide whether 3 side lengths can make a triangle

These skills help you describe triangles correctly and check whether a triangle is even possible.

1. Classifying triangles by side lengths

Triangles can be grouped by how many sides are the same length.

• Equilateral triangle: all 3 sides are equal
• Isosceles triangle: exactly 2 sides are equal
• Scalene triangle: no sides are equal

Examples:

• Sides: \(5, 5, 5\) → equilateral
• Sides: \(7, 7, 4\) → isosceles
• Sides: \(3, 4, 5\) → scalene

A quick way to classify by sides is to look for matches:

• 3 matching sides → equilateral
• 2 matching sides → isosceles
• 0 matching sides → scalene

2. Classifying triangles by angle size

Triangles can also be grouped by the sizes of their angles.

• Acute triangle: all 3 angles are less than \(90^\circ\)
• Right triangle: one angle is exactly \(90^\circ\)
• Obtuse triangle: one angle is greater than \(90^\circ\)

Remember:

• An acute angle is less than \(90^\circ\).
• A right angle is exactly \(90^\circ\).
• An obtuse angle is greater than \(90^\circ\).

Examples:

• Angles: \(60^\circ, 60^\circ, 60^\circ\) → acute
• Angles: \(30^\circ, 60^\circ, 90^\circ\) → right
• Angles: \(100^\circ, 40^\circ, 40^\circ\) → obtuse

Important fact: the angles in any triangle always add up to \(180^\circ\).

That means if you know two angles, you can find the third angle.

For example, if a triangle has angles \(50^\circ\) and \(60^\circ\), then the third angle is

$$180^\circ - 50^\circ - 60^\circ = 70^\circ$$

So the triangle is acute, because all three angles are less than \(90^\circ\).

3. A triangle can have two names

A triangle can be classified by sides and by angles at the same time.

For example:

• A triangle with sides \(5, 5, 8\) is isosceles by sides.
• If one of its angles is greater than \(90^\circ\), it is also obtuse by angles.

So one triangle might be called an isosceles obtuse triangle or a scalene right triangle.

4. The Triangle Inequality Theorem

Not every set of 3 numbers can make a triangle. The side lengths must follow a rule called the Triangle Inequality Theorem.

Triangle Inequality Theorem: In any triangle, the sum of any two side lengths must be greater than the third side length.

If the side lengths are \(a\), \(b\), and \(c\), then all three of these must be true:

$$a+b>c$$ $$a+c>b$$ $$b+c>a$$

The key words are greater than. If the sum is equal to the third side, that does not make a triangle.

Why does this rule make sense?

Imagine trying to connect two short sides to reach across one very long side. If the two short sides together are not longer than the long side, they cannot bend enough to make a closed shape. So no triangle can form.

5. An easier way to check

A quick strategy is to look at the longest side. If the sum of the two shorter sides is greater than the longest side, then the other two checks will also work.

For example, with side lengths \(4\), \(6\), and \(9\):

• Longest side is \(9\)
• Check: \(4+6=10\)
• Since \(10>9\), these side lengths can make a triangle

But if the side lengths are \(2\), \(3\), and \(5\):

• Longest side is \(5\)
• Check: \(2+3=5\)
• Since \(5\) is not greater than \(5\), these side lengths cannot make a triangle

6. Worked Examples

Example 1: Classify a triangle by sides

A triangle has side lengths \(8\), \(8\), and \(5\). What type of triangle is it by sides?

Step 1: Look for equal side lengths.

• Two sides are \(8\) and \(8\)
• One side is \(5\)

Step 2: Decide the type.

Because exactly two sides are equal, this is an isosceles triangle.

Example 2: Classify a triangle by angles

A triangle has angles \(90^\circ\), \(55^\circ\), and \(35^\circ\). What type of triangle is it by angles?

Step 1: Look for a right angle.

• One angle is \(90^\circ\)

Step 2: Decide the type.

If a triangle has one angle of \(90^\circ\), it is a right triangle.

Example 3: Find the missing angle and classify

A triangle has angles \(45^\circ\) and \(45^\circ\). Find the third angle and classify the triangle by angles.

Step 1: Use the fact that triangle angles add to \(180^\circ\).

$$180^\circ - 45^\circ - 45^\circ = 90^\circ$$

Step 2: Identify the triangle type.

Since one angle is \(90^\circ\), the triangle is a right triangle.

Extra note: Because two angles are equal, the triangle is also isosceles by sides. So it is an isosceles right triangle.

Example 4: Use the Triangle Inequality Theorem

Can side lengths \(3\), \(4\), and \(8\) make a triangle?

Step 1: Find the longest side.

• The longest side is \(8\)

Step 2: Add the two shorter sides.

$$3+4=7$$

Step 3: Compare the sum to the longest side.

• Is \(7>8\)? No.

Since the sum of the two shorter sides is not greater than the longest side, these lengths cannot make a triangle.

7. Common mistakes to avoid

• Mixing up side names and angle names
  Equilateral, isosceles, and scalene describe sides. Acute, right, and obtuse describe angles.
• Forgetting that the angle sum is \(180^\circ\)
  Always use \(180^\circ\) to find a missing angle in a triangle.
• Using equal to instead of greater than
  In the Triangle Inequality Theorem, the sum must be greater than the third side, not just equal to it.
• Only checking whether the numbers look close
  Always do the addition to be sure.

8. Quick practice ideas

• Sides \(6,6,6\) → equilateral
• Sides \(4,4,7\) → isosceles
• Sides \(2,5,9\) → not a triangle, because \(2+5=7\) and \(7\) is not greater than \(9\)
• Angles \(80^\circ, 70^\circ, 30^\circ\) → acute
• Angles \(110^\circ, 40^\circ, 30^\circ\) → obtuse

Summary

Triangles can be classified in two ways. By sides, they are equilateral, isosceles, or scalene. By angles, they are acute, right, or obtuse.

Also, three side lengths can only form a triangle if the sum of any two side lengths is greater than the third side length. This is the Triangle Inequality Theorem.

If you remember these rules, you will be able to describe triangles correctly and decide whether a triangle is possible.

Lesson: The Sum of Interior Angles in a Triangle

Triangles are one of the most important shapes in geometry. No matter what kind of triangle you draw, it always has 3 sides, 3 vertices, and 3 interior angles.

In this lesson, you will learn a very important rule: the interior angles in any triangle always add up to 180 degrees.

This means that if you know two angles in a triangle, you can figure out the third one.

What are interior angles?

Interior angles are the angles inside a shape. In a triangle, the three corners each make one interior angle.

If a triangle has angles labeled \(A\), \(B\), and \(C\), then the rule is:

$$A + B + C = 180^\circ$$

This is true for every triangle:

• small triangles
• large triangles
• skinny triangles
• wide triangles
• right triangles
• acute triangles
• obtuse triangles

Why is the sum always 180 degrees?

A straight line measures \(180^\circ\). If you look at the three interior angles of a triangle in the right way, they fit together to make a straight angle. That is why their total is always \(180^\circ\).

You do not have to prove this every time, but it is important to remember the rule:

$$\text{Angle 1} + \text{Angle 2} + \text{Angle 3} = 180^\circ$$

How to find a missing angle

If you know two angles in a triangle, you can find the missing angle in two steps:

1. Add the two known angles.
2. Subtract that sum from \(180^\circ\).

In symbols:

$$\text{Missing angle} = 180^\circ - (\text{angle 1} + \text{angle 2})$$

Worked Example 1

A triangle has angles of \(50^\circ\) and \(60^\circ\). Find the third angle.

Step 1: Add the known angles.

$$50^\circ + 60^\circ = 110^\circ$$

Step 2: Subtract from \(180^\circ\).

$$180^\circ - 110^\circ = 70^\circ$$

Answer: The third angle is \(70^\circ\).

Check:

$$50^\circ + 60^\circ + 70^\circ = 180^\circ$$

Worked Example 2

A triangle has angles of \(90^\circ\) and \(35^\circ\). Find the third angle.

Step 1: Add the known angles.

$$90^\circ + 35^\circ = 125^\circ$$

Step 2: Subtract from \(180^\circ\).

$$180^\circ - 125^\circ = 55^\circ$$

Answer: The third angle is \(55^\circ\).

Check:

$$90^\circ + 35^\circ + 55^\circ = 180^\circ$$

This triangle is a right triangle because one angle is \(90^\circ\). Even in a right triangle, the angle sum is still \(180^\circ\).

Worked Example 3

A triangle has angles of \(x\), \(65^\circ\), and \(75^\circ\). Find \(x\).

Use the triangle angle sum rule:

$$x + 65^\circ + 75^\circ = 180^\circ$$

Add the known angles:

$$65^\circ + 75^\circ = 140^\circ$$

Now solve for \(x\):

$$x = 180^\circ - 140^\circ$$ $$x = 40^\circ$$

Answer: \(x = 40^\circ\)

Worked Example 4

A triangle has two equal angles. One of them is \(48^\circ\). Find all three angles.

If two angles are equal and one is \(48^\circ\), then the other equal angle is also \(48^\circ\).

Add those two angles:

$$48^\circ + 48^\circ = 96^\circ$$

Subtract from \(180^\circ\) to find the third angle:

$$180^\circ - 96^\circ = 84^\circ$$

Answer: The three angles are \(48^\circ\), \(48^\circ\), and \(84^\circ\).

Important ideas to remember

• The angles inside a triangle are called interior angles.
• The sum of the interior angles in any triangle is always \(180^\circ\).
• To find a missing angle, subtract the sum of the known angles from \(180^\circ\).
• Your final three angles should add up to \(180^\circ\).

Common mistakes

• Forgetting to subtract from 180: First add the known angles, then subtract.
• Adding incorrectly: Be careful when adding the two known angles.
• Using an answer that is too large: A missing angle in a triangle must make the total exactly \(180^\circ\).
• Mixing up interior and outside angles: This lesson is about angles inside the triangle.

Quick practice

1. A triangle has angles \(40^\circ\) and \(80^\circ\). What is the third angle?
2. A triangle has angles \(90^\circ\) and \(40^\circ\). What is the third angle?
3. A triangle has angles \(x\), \(30^\circ\), and \(100^\circ\). Find \(x\).

Answers to quick practice

1. $$180^\circ - (40^\circ + 80^\circ) = 60^\circ$$
2. $$180^\circ - (90^\circ + 40^\circ) = 50^\circ$$
3. $$x = 180^\circ - (30^\circ + 100^\circ) = 50^\circ$$

Summary

The three interior angles of any triangle always add up to \(180^\circ\). This rule works for every type of triangle.

When you know two angles, add them together and subtract from \(180^\circ\) to find the missing angle. Always check by making sure all three angles total \(180^\circ\).

Quadrilateral Properties and Hierarchy

A quadrilateral is a polygon with 4 sides, 4 angles, and 4 vertices.

In this lesson, you will learn how to identify different kinds of quadrilaterals by looking at their sides and angles. You will also learn how these shapes fit into a hierarchy, which means an organized family tree of shapes.

When classifying quadrilaterals, we pay attention to these important properties:

• Parallel sides: lines that never meet
• Congruent sides: sides that have the same length
• Right angles: angles that measure \(90^\circ\)

One useful fact is that the sum of the inside angles of any quadrilateral is:

$$360^\circ$$

This fact helps us check whether angle measures in a quadrilateral make sense.

1. The general quadrilateral

A general quadrilateral is any 4-sided shape. It does not need to have parallel sides, equal sides, or right angles.

So, every square is a quadrilateral, every rectangle is a quadrilateral, and every rhombus is a quadrilateral. But not every quadrilateral is one of those special shapes.

2. Trapezoid

A trapezoid is a quadrilateral with at least one pair of parallel sides.

The parallel sides are called the bases. The other two sides are called the legs.

Example of what to look for:

• 1 pair of parallel sides → trapezoid
• The other pair does not need to be parallel

3. Parallelogram

A parallelogram is a quadrilateral with 2 pairs of parallel sides.

Parallelograms have several important properties:

• Opposite sides are parallel
• Opposite sides are congruent
• Opposite angles are equal

This means if one side is 8 cm, the opposite side is also 8 cm. If one angle is \(70^\circ\), the opposite angle is also \(70^\circ\).

4. Rectangle

A rectangle is a special kind of parallelogram.

A rectangle has:

• 2 pairs of parallel sides
• 4 right angles

Since a rectangle is a parallelogram, it also has congruent opposite sides.

Important idea: Every rectangle is a parallelogram, but not every parallelogram is a rectangle.

5. Rhombus

A rhombus is also a special kind of parallelogram.

A rhombus has:

• 2 pairs of parallel sides
• 4 congruent sides

The angles in a rhombus do not have to be right angles. Some rhombuses look slanted.

Important idea: Every rhombus is a parallelogram, but not every parallelogram is a rhombus.

6. Square

A square is the most special quadrilateral in this lesson because it belongs to more than one group.

A square has:

• 2 pairs of parallel sides
• 4 congruent sides
• 4 right angles

Because of these properties, a square is:

• a quadrilateral
• a parallelogram
• a rectangle
• a rhombus

This is an important part of the hierarchy. A square fits in all of those groups because it has all of their properties.

The hierarchy of quadrilaterals

Here is one way to organize these shapes from general to more special:

• Quadrilateral
  • Trapezoid
  • Parallelogram
    • Rectangle
    • Rhombus
    • Square (both a rectangle and a rhombus)

You can think of this as a family tree. The more properties a shape has, the more special it becomes.

How to classify a quadrilateral

When you see a quadrilateral, ask these questions in order:

1. Does it have 4 sides? If yes, it is a quadrilateral.
2. Does it have any parallel sides?
3. Does it have 2 pairs of parallel sides? If yes, it is a parallelogram.
4. Does it have 4 right angles? If yes, it is a rectangle.
5. Does it have 4 equal sides? If yes, it is a rhombus.
6. Does it have both 4 right angles and 4 equal sides? If yes, it is a square.

Worked Example 1: Classifying by parallel sides

A shape has 4 sides. One pair of opposite sides is parallel, but the other pair is not parallel.

Step 1: It has 4 sides, so it is a quadrilateral.

Step 2: It has at least one pair of parallel sides.

Conclusion: The shape is a trapezoid.

Worked Example 2: Classifying a parallelogram

A quadrilateral has 2 pairs of parallel sides. Its side lengths are 6 cm, 9 cm, 6 cm, and 9 cm. It does not have right angles.

Step 1: With 2 pairs of parallel sides, it is a parallelogram.

Step 2: The sides are not all equal, so it is not a rhombus.

Step 3: It does not have right angles, so it is not a rectangle.

Conclusion: The most specific name is parallelogram.

Worked Example 3: Rectangle or square?

A quadrilateral has 4 right angles and side lengths 5 cm, 8 cm, 5 cm, and 8 cm.

Step 1: It has 2 pairs of equal opposite sides.

Step 2: It has 4 right angles, so it is a rectangle.

Step 3: Not all 4 sides are equal, so it is not a square.

Conclusion: The shape is a rectangle.

Worked Example 4: Using angles to classify

A quadrilateral has 4 congruent sides. One angle measures \(90^\circ\).

Step 1: Four congruent sides means the shape is a rhombus.

Step 2: One angle is \(90^\circ\). In a parallelogram, if one angle is a right angle, then all angles are right angles.

Step 3: So this rhombus also has 4 right angles.

Conclusion: The shape is a square.

Using the angle sum of a quadrilateral

Remember that the angle sum is:

$$360^\circ$$

If three angles of a quadrilateral are \(90^\circ\), \(90^\circ\), and \(100^\circ\), then the fourth angle is:

$$360^\circ - (90^\circ + 90^\circ + 100^\circ) = 80^\circ$$

This helps you check the shape. Since not all four angles are right angles, the quadrilateral cannot be a rectangle or a square.

Common mistakes to avoid

• Thinking a square is not a rectangle. A square is a rectangle because it has 4 right angles.
• Thinking a square is not a rhombus. A square is a rhombus because it has 4 equal sides.
• Forgetting that a rectangle must have 4 right angles, not just one.
• Forgetting that a parallelogram must have 2 pairs of parallel sides.
• Mixing up equal sides and parallel sides. These are different properties.

Quick comparison chart

• Quadrilateral: any 4-sided shape
• Trapezoid: at least 1 pair of parallel sides
• Parallelogram: 2 pairs of parallel sides
• Rectangle: a parallelogram with 4 right angles
• Rhombus: a parallelogram with 4 congruent sides
• Square: a parallelogram with 4 congruent sides and 4 right angles

Summary

Quadrilaterals are 4-sided shapes. We classify them by checking parallel sides, equal side lengths, and angle measures.

The hierarchy helps us see that some shapes belong to more than one group. For example, every square is both a rectangle and a rhombus, and both of those are parallelograms.

If you classify carefully from general to special, you can name each quadrilateral correctly.

Area of Rectangles and Parallelograms

When we find the area of a shape, we are finding how much surface it covers.

Area is measured in square units, such as square centimeters \\(cm^2\\), square meters \\(m^2\\), or square inches \\(in^2\\).

In this lesson, you will learn how to find the area of rectangles and parallelograms, and why both formulas are based on multiplying a base by a height.

1. Area of a Rectangle

A rectangle has four right angles. To find its area, multiply its length by its width.

The formula is:

$$A = l \times w$$

Some books also use base and height instead of length and width. Then the formula looks like this:

$$A = b \times h$$

This works because a rectangle is made of rows of equal squares.

For example, if a rectangle is 5 units long and 3 units wide, it has 3 rows of 5 squares, or 5 columns of 3 squares. Either way, the total number of square units is:

$$5 \times 3 = 15$$

So the area is \\(15\\) square units.

2. Why Multiplication Works for Rectangles

Imagine a rectangle drawn on grid paper. If it is 4 squares across and 6 squares down, then it contains 6 rows with 4 squares in each row.

You can add them:

$$4 + 4 + 4 + 4 + 4 + 4 = 24$$

Multiplication is a faster way to do the same thing:

$$6 \times 4 = 24$$

So the area is \\(24\\) square units.

3. Area of a Parallelogram

A parallelogram is a four-sided shape with two pairs of opposite sides parallel.

It may look slanted, but you can still find its area using a very similar formula:

$$A = b \times h$$

Here, \\(b\\) is the base, and \\(h\\) is the height.

The height is the straight up-and-down distance from the base to the opposite side. It must be perpendicular to the base, which means it forms a right angle.

Important: The slanted side of a parallelogram is usually not the height.

4. Why the Parallelogram Formula Works

A parallelogram can be changed into a rectangle without changing its area.

Imagine cutting a small triangle off one side of the parallelogram and moving it to the other side. The new shape becomes a rectangle.

The rectangle has the same base and the same height as the parallelogram. Since the area of the rectangle is \\(b \times h\\), the area of the parallelogram is also:

$$A = b \times h$$

This is why both shapes use the same kind of formula.

5. Be Careful About Height

Students often make a mistake by multiplying the base by the slanted side.

For a parallelogram, you must use the perpendicular height, not just any side length.

• Base: the side you choose to measure along the bottom
• Height: the shortest distance straight up to the opposite side
• Slanted side: usually not used for area unless it is also perpendicular

6. Steps for Finding Area

For both rectangles and parallelograms, you can follow these steps:

1. Identify the base.
2. Identify the height.
3. Multiply base by height.
4. Write the answer in square units.

Worked Example 1: Rectangle

Find the area of a rectangle with length 8 cm and width 5 cm.

Use the formula:

$$A = l \times w$$

Substitute the numbers:

$$A = 8 \times 5 = 40$$

The area is 40 \\(cm^2\\).

Worked Example 2: Rectangle on a Grid

A rectangle covers 7 squares across and 4 squares down. What is its area?

Multiply:

$$A = 7 \times 4 = 28$$

The area is 28 square units.

Worked Example 3: Parallelogram

A parallelogram has a base of 9 m and a height of 6 m. Find its area.

Use the formula:

$$A = b \times h$$

Substitute the values:

$$A = 9 \times 6 = 54$$

The area is 54 \\(m^2\\).

Worked Example 4: Watch Out for the Slanted Side

A parallelogram has a base of 10 in, a height of 4 in, and a slanted side of 6 in. What is its area?

Only the base and the perpendicular height are used.

So:

$$A = b \times h = 10 \times 4 = 40$$

The area is 40 \\(in^2\\).

The 6-inch slanted side is not needed.

7. Comparing Rectangle and Parallelogram Area

Suppose a rectangle and a parallelogram both have base 12 units and height 3 units.

Rectangle area:

$$12 \times 3 = 36$$

Parallelogram area:

$$12 \times 3 = 36$$

Even though the shapes look different, they have the same area because they have the same base and height.

8. Common Mistakes to Avoid

• Forgetting to write square units
• Using the slanted side instead of the height in a parallelogram
• Adding side lengths instead of multiplying base and height
• Mixing up area and perimeter

Remember:

• Area tells how much space is inside a shape.
• Perimeter tells the distance around a shape.

9. Quick Practice Ideas

Try these on your own:

• A rectangle with length 11 cm and width 2 cm
• A parallelogram with base 7 m and height 8 m
• A rectangle with base 9 units and height 9 units
• A parallelogram with base 13 in and height 5 in

Use \\(A = b \times h\\) or \\(A = l \times w\\).

Summary

The area of a rectangle is found by multiplying length by width:

$$A = l \times w$$

The area of a parallelogram is found by multiplying base by height:

$$A = b \times h$$

A parallelogram can be rearranged into a rectangle, which shows why the formula works.

Always make sure you use the perpendicular height and write your answer in square units.

Area of Triangles

When we find the area of a shape, we are finding how much space is inside it. Area is measured in square units, such as square centimeters \\(cm^2\\), square meters \\(m^2\\), or square inches \\(in^2\\).

In this lesson, you will learn how to find the area of a triangle and why the triangle area formula works. A triangle’s area is connected to the area of a parallelogram that has the same base and height.

Main Idea: The area of a triangle is exactly half the area of a parallelogram with the same base and height.

If a parallelogram has base \\(b\\) and height \\(h\\), its area is:

$$A = b \times h$$

A diagonal can split that parallelogram into 2 equal triangles. So each triangle has half the area:

$$A = \frac{1}{2}bh$$

This gives us the formula for the area of a triangle:

$$\text{Area of a triangle} = \frac{1}{2} \times \text{base} \times \text{height}$$

Or, using variables:

$$A = \frac{1}{2}bh$$

What do base and height mean?

• Base: any side of the triangle you choose as the bottom.
• Height: the straight up-and-down distance from the base to the opposite vertex.

The height must be perpendicular to the base. That means it meets the base at a right angle, \\(90^\circ\\).

Sometimes the height is inside the triangle. Sometimes it is shown outside the triangle with a dashed line. Either way, the height is always the distance straight from the base to the opposite point.

Why does the formula make sense?

Imagine two copies of the same triangle. If you put them together, they can form a parallelogram. The parallelogram has the same base and the same height as the triangle.

Since the area of the parallelogram is \\(b \times h\\), one triangle must be half of that:

$$\frac{1}{2}(b \times h) = \frac{1}{2}bh$$

This is why triangle area always uses \\(\frac{1}{2}\\).

Steps for finding the area of a triangle

1. Find the base.
2. Find the height that matches that base.
3. Multiply base and height.
4. Take half of the product.
5. Write the answer in square units.

So the process is:

$$A = \frac{1}{2}bh$$

Important reminder: Do not use a slanted side as the height unless it is actually perpendicular to the base.

Worked Example 1: Basic triangle

A triangle has a base of \\(8\\) cm and a height of \\(5\\) cm. Find its area.

Use the formula:

$$A = \frac{1}{2}bh$$

Substitute the values:

$$A = \frac{1}{2}(8)(5)$$ $$A = \frac{1}{2}(40)$$ $$A = 20$$

The area is \\(20\,cm^2\\).

Worked Example 2: Larger numbers

A triangle has a base of \\(14\\) m and a height of \\(9\\) m. Find its area.

$$A = \frac{1}{2}bh$$ $$A = \frac{1}{2}(14)(9)$$ $$A = \frac{1}{2}(126)$$ $$A = 63$$

The area is \\(63\,m^2\\).

Worked Example 3: Understanding the parallelogram connection

A parallelogram has base \\(10\\) cm and height \\(6\\) cm.

First find the area of the parallelogram:

$$A = bh = 10 \times 6 = 60$$

If the parallelogram is cut along a diagonal, it makes 2 equal triangles.

So the area of each triangle is:

$$\frac{60}{2} = 30$$

Each triangle has area \\(30\,cm^2\\).

We can also check with the triangle formula:

$$A = \frac{1}{2}(10)(6) = 30$$

The answer matches.

Worked Example 4: Choosing the correct height

A triangle has a base of \\(12\\) in and a height of \\(7\\) in. Another side is \\(9\\) in. Find the area.

Even though one side is \\(9\\) in, we only use the height that is perpendicular to the base.

So we use base \\(12\\) in and height \\(7\\) in:

$$A = \frac{1}{2}(12)(7)$$ $$A = \frac{1}{2}(84)$$ $$A = 42$$

The area is \\(42\,in^2\\).

Common mistakes to avoid

• Forgetting to divide by 2.
• Using a side length that is not the height.
• Forgetting to write square units.
• Mixing up area and perimeter. Area measures the space inside the triangle, not the distance around it.

Quick check

Try these on your own:

• A triangle has base \\(6\\) cm and height \\(4\\) cm.
• A triangle has base \\(15\\) m and height \\(8\\) m.
• A triangle comes from half of a parallelogram with area \\(50\\,ft^2\\). What is the triangle’s area?

Answers:

• \\(A = \frac{1}{2}(6)(4) = 12\,cm^2\\)
• \\(A = \frac{1}{2}(15)(8) = 60\,m^2\\)
• \\(25\,ft^2\\)

Summary

The area of a triangle is found with the formula:

$$A = \frac{1}{2}bh$$

This formula works because a triangle is half of a parallelogram with the same base and height. To find area correctly, always use a base and the matching perpendicular height, then remember to divide by 2 and write square units.

Lesson: Area of Trapezoids and Composite Polygons

In geometry, area means the amount of space inside a flat shape. We measure area in square units, such as square centimeters cm^2, square meters m^2, or square inches in^2.

In this lesson, you will learn how to find the area of a trapezoid and how to find the area of a composite polygon. A composite polygon is a shape made by joining simpler shapes, like rectangles and triangles.

These skills are useful because many real shapes are not just one simple rectangle or triangle. If you can break a shape into parts, you can find its total area.

1. What is a trapezoid?

A trapezoid is a quadrilateral with one pair of parallel sides. The parallel sides are called the bases. The distance straight up and down between the bases is called the height.

Be careful: the height must be the perpendicular distance between the bases. It is not usually one of the slanted sides.

Formula for the area of a trapezoid

To find the area of a trapezoid, add the two bases, multiply by the height, and then divide by 2.

$$A = \frac{(b_1 + b_2)h}{2}$$

Here:

• \(b_1\) and \(b_2\) are the lengths of the two bases
• \(h\) is the height
• \(A\) is the area

You can also think of this formula as:

$$A = \left(\frac{b_1+b_2}{2}\right)h$$

This means: find the average of the two bases, then multiply by the height.

Why does the trapezoid formula make sense?

A trapezoid can be split into simpler shapes, like a rectangle and triangles. When those parts are added together, the formula becomes:

$$A = \frac{(b_1 + b_2)h}{2}$$

So even though the formula may look new, it comes from shapes you already know.

2. Steps for finding the area of a trapezoid

1. Identify the two parallel bases.
2. Find the height, the straight distance between the bases.
3. Substitute the values into the formula.
4. Simplify carefully.
5. Write the answer in square units.

Worked Example 1: Find the area of a trapezoid

A trapezoid has bases of 8 cm and 14 cm. Its height is 5 cm. Find the area.

Step 1: Write the formula.

$$A = \frac{(b_1+b_2)h}{2}$$

Step 2: Substitute the numbers.

$$A = \frac{(8+14)(5)}{2}$$

Step 3: Add inside the parentheses.

$$A = \frac{22\cdot 5}{2}$$

Step 4: Multiply.

$$A = \frac{110}{2}$$

Step 5: Divide.

$$A = 55$$

Answer: The area is 55 \(cm^2\).

Worked Example 2: Watch out for the height

A trapezoid has bases of 12 m and 18 m. One slanted side is 7 m, and the height is 6 m. Find the area.

The slanted side is not the height. Use the perpendicular height of 6 m.

$$A = \frac{(12+18)(6)}{2}$$ $$A = \frac{30\cdot 6}{2}$$ $$A = \frac{180}{2}$$ $$A = 90$$

Answer: The area is 90 \(m^2\).

3. What is a composite polygon?

A composite polygon is a larger shape made from two or more smaller shapes. For example, a shape might be made from:

• a rectangle and a triangle,
• two rectangles,
• a trapezoid and a rectangle.

To find the area of a composite polygon, you usually break it into simpler shapes whose areas you already know.

Helpful area formulas to remember

• Rectangle: $$A = lw$$
• Triangle: $$A = \frac{1}{2}bh$$
• Trapezoid: $$A = \frac{(b_1+b_2)h}{2}$$

4. Steps for finding the area of a composite polygon

1. Look at the shape carefully.
2. Split it into familiar shapes, such as rectangles, triangles, or trapezoids.
3. Find the area of each smaller shape.
4. Add the areas together.
5. Write the final answer in square units.

Sometimes a shape may have a missing corner or cut-out part. In that case, you can:

• find the area of a larger simple shape, then
• subtract the area of the missing part.

Worked Example 3: Composite polygon made of two rectangles

An L-shaped figure is made of two rectangles:

• Rectangle 1: length 10 cm and width 4 cm
• Rectangle 2: length 6 cm and width 3 cm

Find the total area.

Step 1: Find the area of Rectangle 1.

$$A_1 = 10\cdot 4 = 40$$

Step 2: Find the area of Rectangle 2.

$$A_2 = 6\cdot 3 = 18$$

Step 3: Add the areas.

$$A_{total} = 40 + 18 = 58$$

Answer: The total area is 58 \(cm^2\).

Worked Example 4: Composite polygon made of a rectangle and a triangle

A shape looks like a house. It has:

• a rectangle with width 8 m and height 5 m
• a triangle on top with base 8 m and height 3 m

Find the total area.

Step 1: Find the area of the rectangle.

$$A_{rectangle} = 8\cdot 5 = 40$$

Step 2: Find the area of the triangle.

$$A_{triangle} = \frac{1}{2}(8)(3) = 12$$

Step 3: Add the areas.

$$A_{total} = 40 + 12 = 52$$

Answer: The total area is 52 \(m^2\).

5. Using subtraction with composite polygons

Sometimes a composite shape is easier to solve by thinking of it as a large rectangle with a smaller piece missing.

For example, imagine a large rectangle measuring 12 cm by 9 cm, with a small rectangle cut out that measures 4 cm by 2 cm.

Step 1: Find the area of the large rectangle.

$$A_{large} = 12\cdot 9 = 108$$

Step 2: Find the area of the cut-out rectangle.

$$A_{cutout} = 4\cdot 2 = 8$$

Step 3: Subtract.

$$A_{remaining} = 108 - 8 = 100$$

Answer: The area of the composite shape is 100 \(cm^2\).

6. Tips for success

• Always check the height. For trapezoids and triangles, the height is the straight up-and-down distance, not a slanted side.
• Label parts clearly. Writing the lengths on a sketch can help you see which formula to use.
• Split shapes in a smart way. Try to make rectangles, triangles, and trapezoids.
• Use square units. Area is always measured in square units.
• Add or subtract carefully. Composite shapes often need more than one step.

7. Common mistakes to avoid

• Using a slanted side as the height of a trapezoid
• Forgetting to divide by 2 in the trapezoid formula
• Adding side lengths instead of finding area
• Forgetting to include all parts of a composite figure
• Leaving off the square units

8. Quick check for understanding

Try these on your own:

• A trapezoid has bases 9 cm and 15 cm and height 4 cm. What is its area?
• A composite figure is made of a rectangle measuring 7 m by 5 m and a triangle with base 7 m and height 2 m. What is the total area?

Answers:

1. $$A = \frac{(9+15)(4)}{2} = \frac{24\cdot 4}{2} = 48$$ so the area is 48 \(cm^2\).

2. Rectangle: $$7\cdot 5 = 35$$ Triangle: $$\frac{1}{2}(7)(2) = 7$$ Total: $$35+7=42$$ so the area is 42 \(m^2\).

Summary

The area of a trapezoid is found with the formula $$A = \frac{(b_1+b_2)h}{2}$$, where the bases are the parallel sides and the height is the perpendicular distance between them.

To find the area of a composite polygon, break the figure into smaller shapes such as rectangles, triangles, and trapezoids. Find each area, then add them together, or subtract if part of the shape is missing.

When solving area problems, work carefully, use the correct height, and always write your answer in square units.

Parts of a Circle and Pi (π)

Circles are all around us: wheels, coins, clocks, and plates are all shaped like circles. In this lesson, you will learn the important parts of a circle and understand what pi, written as (\pi), means.

By the end of the lesson, you should be able to name the radius, diameter, and chord of a circle, and explain why (\pi) is special for every circle.

1. What is a circle?

A circle is a round shape made of all the points that are the same distance from one fixed point. That fixed point is called the center.

If you pick any point on the edge of the circle, it will be the same distance from the center as every other point on the edge.

2. Important parts of a circle

To understand circles, we need to know the names of some important parts.

• Center: the point exactly in the middle of the circle.
• Radius: a line segment from the center to any point on the circle.
• Diameter: a line segment that goes across the circle through the center, with both ends on the circle.
• Chord: a line segment with both ends on the circle.
• Circumference: the distance all the way around the circle.

3. Radius and diameter

The radius and diameter are closely connected.

A diameter is made of two radii placed end to end through the center. Because of this, the diameter is always twice the radius.

We can write this relationship as:

$$d = 2r$$

And if you know the diameter, you can find the radius by dividing by 2:

$$r = \frac{d}{2}$$

Here, (r) stands for radius and (d) stands for diameter.

4. What is a chord?

A chord is any line segment that connects two points on the circle.

The diameter is actually a very special chord because it goes through the center. This means:

• Every diameter is a chord.
• Not every chord is a diameter.

A chord can be short or long. The longest chord in a circle is the diameter.

5. What is circumference?

The circumference is the distance around a circle. It is like the perimeter of a polygon, but since a circle has no straight sides, we use the special word circumference.

If you wrapped a string all the way around a can and then measured the string, that length would be the circumference of the circle on the top of the can.

6. Discovering pi, (\pi)

Now for the special number pi, written as (\pi).

For every circle, if you divide the circumference by the diameter, you always get the same number. That number is pi.

In math symbols:

$$\pi = \frac{C}{d}$$

Here, (C) means circumference and (d) means diameter.

This is amazing because it works for small circles, large circles, and every circle in between.

Pi is approximately:

$$\pi \approx 3.14$$

This means the circumference is a little more than 3 times the diameter.

For example, if a circle has diameter 1 unit, then its circumference is about 3.14 units. If a circle has diameter 10 units, then its circumference is about 31.4 units.

7. Circumference formulas using pi

Since

$$\pi = \frac{C}{d}$$

we can multiply both sides by (d) to get a formula for circumference:

$$C = \pi d$$

Because the diameter is twice the radius, (d = 2r), we can also write:

$$C = 2\pi r$$

Both formulas mean the same thing. Use whichever one matches the information you are given.

• If you know the diameter, use (C = \pi d).
• If you know the radius, use (C = 2\pi r).

8. Worked Examples

Example 1: Find the diameter when the radius is 6 cm.

Use the relationship:

$$d = 2r$$

Substitute (r = 6):

$$d = 2(6) = 12$$

Answer: The diameter is 12 cm.

Example 2: Find the radius when the diameter is 18 m.

Use the relationship:

$$r = \frac{d}{2}$$

Substitute (d = 18):

$$r = \frac{18}{2} = 9$$

Answer: The radius is 9 m.

Example 3: Find the circumference when the diameter is 10 in. Use (\pi \approx 3.14).

Use the formula:

$$C = \pi d$$

Substitute (d = 10) and (\pi \approx 3.14):

$$C \approx 3.14 \times 10 = 31.4$$

Answer: The circumference is about 31.4 inches.

Example 4: Find the circumference when the radius is 7 cm. Use (\pi \approx 3.14).

Use the formula:

$$C = 2\pi r$$

Substitute (r = 7):

$$C \approx 2 \times 3.14 \times 7$$ $$C \approx 6.28 \times 7 = 43.96$$

Answer: The circumference is about 43.96 cm.

9. How to tell the parts apart

• A radius goes from the center to the circle.
• A diameter goes across the whole circle and passes through the center.
• A chord connects two points on the circle, but it does not have to go through the center.

One easy way to remember is:

• Radius = from the middle to the rim
• Diameter = all the way across through the middle
• Chord = across the circle from one edge to another edge

10. Common mistakes to avoid

• Mixing up radius and diameter: the diameter is always twice the radius.
• Thinking every chord is a diameter: only the chord that goes through the center is a diameter.
• Using the wrong formula: make sure you know whether you are given radius or diameter.
• Forgetting that (\pi) is about 3.14: when asked for a decimal answer, use (\pi \approx 3.14) unless your teacher says otherwise.

11. Quick review

1. The center is the middle of the circle.
2. The radius goes from the center to the circle.
3. The diameter goes across the circle through the center.
4. A chord connects two points on the circle.
5. The circumference is the distance around the circle.
6. (\pi) is the constant ratio of circumference to diameter:

$$\pi = \frac{C}{d}$$

And the circumference formulas are:

$$C = \pi d$$ $$C = 2\pi r$$

Summary

A circle has important parts called the radius, diameter, and chord. The diameter is twice the radius, and it is also the longest chord.

The distance around a circle is called the circumference. Pi, written as (\pi), is the number you get when you divide any circle's circumference by its diameter. This is why we use (C = \pi d) or (C = 2\pi r) to find circumference.

Area of Triangles and Parallelograms

In geometry, area means the amount of space inside a flat shape. We measure area in square units, such as square centimeters, square inches, or square meters.

You may already know how to find the area of a rectangle. For a rectangle, we multiply length times width:

$$\text{Area of a rectangle} = \text{length} \times \text{width}$$

In this lesson, you will learn how to find the area of parallelograms and triangles. We will connect both of these shapes to rectangles, so the formulas make sense and are easier to remember.

1. What is a parallelogram?

A parallelogram is a four-sided shape with two pairs of opposite sides that are parallel. Its opposite sides are the same length.

To find its area, we use the shape's base and height.

• The base is one side you choose to be the bottom.
• The height is the straight up-and-down distance from the base to the opposite side.

The height must make a right angle with the base. It is not usually the slanted side.

Important: In a parallelogram, the slanted side is often longer than the height. Do not confuse the slanted side with the height.

2. Why does the parallelogram formula work?

A parallelogram can be changed into a rectangle without changing its area. Imagine cutting off a triangular piece from one side and moving it to the other side. The new shape becomes a rectangle.

The rectangle has the same base and the same height as the parallelogram. Since the area of a rectangle is base times height, the area of a parallelogram is also base times height.

$$\text{Area of a parallelogram} = b \times h$$

Here, \(b\) means base and \(h\) means height.

3. What is a triangle?

A triangle is a three-sided shape. Just like with parallelograms, we find its area using a base and a height.

The height is the shortest distance from the base to the opposite corner. It must meet the base at a right angle.

4. Why does the triangle formula work?

If you put two matching triangles together, they can make a parallelogram. That means one triangle is half of that parallelogram.

Since the area of the parallelogram is \(b \times h\), the area of one triangle is half of that:

$$\text{Area of a triangle} = \frac{1}{2} \times b \times h$$

You can also write it as:

$$A = \frac{bh}{2}$$

5. Steps for finding area

For both shapes, start by finding the base and the height. Make sure the height is the distance straight up from the base, not a slanted side unless it makes a right angle.

For a parallelogram:

1. Find the base.
2. Find the height.
3. Multiply: \(b \times h\).
4. Write the answer in square units.

For a triangle:

1. Find the base.
2. Find the height.
3. Multiply: \(b \times h\).
4. Divide by 2.
5. Write the answer in square units.

6. Worked Examples

Example 1: Area of a parallelogram

A parallelogram has a base of 8 cm and a height of 5 cm. Find its area.

Use the formula:

$$A = b \times h$$

Substitute the numbers:

$$A = 8 \times 5 = 40$$

The area is 40 square centimeters, or \(40\text{ cm}^2\).

Example 2: Watch out for the slanted side

A parallelogram has a base of 10 m, a height of 6 m, and a slanted side of 7 m. Find its area.

We use the base and the height, not the slanted side.

$$A = b \times h$$

$$A = 10 \times 6 = 60$$

The area is 60 square meters, or \(60\text{ m}^2\).

Example 3: Area of a triangle

A triangle has a base of 12 in and a height of 4 in. Find its area.

Use the formula:

$$A = \frac{1}{2} \times b \times h$$

Substitute the numbers:

$$A = \frac{1}{2} \times 12 \times 4$$

First multiply \(12 \times 4 = 48\).

Then divide by 2:

$$A = \frac{48}{2} = 24$$

The area is 24 square inches, or \(24\text{ in}^2\).

Example 4: A triangle from a parallelogram idea

A triangle has a base of 9 cm and a height of 10 cm. Find its area.

$$A = \frac{1}{2} \times b \times h$$

$$A = \frac{1}{2} \times 9 \times 10$$

Multiply first:

$$9 \times 10 = 90$$

Now divide by 2:

$$A = \frac{90}{2} = 45$$

The area is 45 square centimeters, or \(45\text{ cm}^2\).

7. Comparing the formulas

• Parallelogram: $$A = b \times h$$
• Triangle: $$A = \frac{1}{2} \times b \times h$$

The only difference is that a triangle is half of a matching parallelogram, so we divide by 2.

8. Common mistakes to avoid

• Do not use the slanted side as the height unless it forms a right angle with the base.
• Do not forget to divide by 2 when finding the area of a triangle.
• Do not forget to write square units, like \(\text{cm}^2\) or \(\text{m}^2\).
• Make sure you multiply the correct base and height that go together.

9. Quick practice thinking

If a parallelogram has base 7 and height 3, its area is:

$$7 \times 3 = 21$$

If a triangle has the same base 7 and height 3, its area is half of 21:

$$\frac{21}{2} = 10.5$$

So the triangle's area is \(10.5\) square units.

This shows clearly that a triangle with the same base and height has half the area of a parallelogram.

10. Summary

Area tells us how much space is inside a shape. To find the area of a parallelogram, multiply the base by the height:

$$A = b \times h$$

To find the area of a triangle, multiply the base by the height and divide by 2:

$$A = \frac{1}{2} \times b \times h$$

These formulas make sense because a parallelogram can be rearranged into a rectangle, and a triangle is half of a matching parallelogram. Always use the straight-up height, not just any side length, and always label your answer with square units.

Circumference of a Circle

Have you ever traced your finger around the edge of a coin, plate, or wheel? The distance all the way around a circle is called its circumference.

Circumference is like the perimeter of a circle. Since a circle has no straight sides, we use a special formula to find the distance around it.

To understand circumference, we first need to know two important parts of a circle.

• Radius: the distance from the center of the circle to the edge.
• Diameter: the distance across the circle through the center.

The diameter is always twice the radius.

$$d = 2r$$

This also means the radius is half the diameter.

$$r = \frac{d}{2}$$

Another very important number for circles is pi, written as \(\pi\). Pi is a number that helps us connect the diameter of a circle to its circumference.

For 6th Grade math, we often use:

• \(\pi \approx 3.14\)
• Sometimes \(\pi \approx \frac{22}{7}\)

The two formulas for circumference are:

$$C = \pi d$$

and

$$C = 2\pi r$$

These formulas mean:

• If you know the diameter, use \(C = \pi d\).
• If you know the radius, use \(C = 2\pi r\).

Both formulas give the same answer because the diameter is twice the radius.

How to find circumference

1. Read the problem carefully.
2. Decide whether you know the radius or diameter.
3. Choose the correct formula.
4. Substitute the number into the formula.
5. Multiply.
6. Write the correct unit, such as cm, m, or in.

Worked Example 1: Using the diameter

A circle has a diameter of 8 cm. Find its circumference.

Use the formula:

$$C = \pi d$$

Substitute \(d = 8\):

$$C = 3.14 \times 8$$

$$C = 25.12$$

So, the circumference is 25.12 cm.

Worked Example 2: Using the radius

A circle has a radius of 5 m. Find its circumference.

Use the formula:

$$C = 2\pi r$$

Substitute \(r = 5\):

$$C = 2 \times 3.14 \times 5$$

$$C = 31.4$$

So, the circumference is 31.4 m.

Worked Example 3: Radius given, but first think carefully

A circular garden has a radius of 7 ft. What is the distance around the garden?

The distance around means the circumference.

Use:

$$C = 2\pi r$$

Substitute \(r = 7\):

$$C = 2 \times 3.14 \times 7$$

$$C = 43.96$$

So, the circumference is 43.96 ft.

Worked Example 4: Diameter is not given directly

A circular clock has a radius of 6 in. Find the circumference.

You can solve this in two ways.

Method 1: Use the radius formula

$$C = 2\pi r$$

$$C = 2 \times 3.14 \times 6$$

$$C = 37.68$$

Method 2: Find the diameter first

$$d = 2r = 2 \times 6 = 12$$

Now use:

$$C = \pi d$$

$$C = 3.14 \times 12 = 37.68$$

Both methods give the same answer, so the circumference is 37.68 in.

Important things to remember

• Circumference means the distance around a circle.
• Radius goes from the center to the edge.
• Diameter goes across the circle through the center.
• The diameter is twice the radius.
• Use \(C = \pi d\) when you know the diameter.
• Use \(C = 2\pi r\) when you know the radius.

Common mistakes to avoid

• Mixing up radius and diameter: If the problem gives the radius, do not use it as the diameter.
• Forgetting the 2 in \(2\pi r\): The formula needs the 2 when using radius.
• Leaving off units: Always write cm, m, ft, in, or the correct unit.
• Using area instead of circumference: Circumference is around the circle, not inside it.

Quick check

If a circle has a diameter of 10 cm, then:

$$C = \pi d = 3.14 \times 10 = 31.4$$

So the circumference is 31.4 cm.

If a circle has a radius of 10 cm, then:

$$C = 2\pi r = 2 \times 3.14 \times 10 = 62.8$$

So the circumference is 62.8 cm.

Notice that a radius of 10 cm makes a bigger circle than a diameter of 10 cm. That is why the answers are different.

Summary

The circumference of a circle is the distance around it. To find circumference, use \(C = \pi d\) if you know the diameter, or use \(C = 2\pi r\) if you know the radius. Always check whether the number given is a radius or diameter, and remember to include units in your answer.

Area of a Circle

When we talk about the area of a shape, we mean how much space is inside it.

For a circle, the area tells us how much flat surface the circle covers. This is different from the circumference, which is the distance around the circle.

In this lesson, you will learn what the parts of a circle are, how to use the area formula, and how to solve area of a circle problems step by step.

1. Parts of a Circle

Before finding the area, it is important to know two circle measurements:

• Radius: the distance from the center of the circle to the edge
• Diameter: the distance across the circle through the center

The radius and diameter are related by this rule:

$$d = 2r$$

This also means:

$$r = \frac{d}{2}$$

2. The Formula for Area

The formula for the area of a circle is:

$$A = \pi r^2$$

In this formula:

• A means area
• r means radius
• \(\pi\) is a special number that is about 3.14
• \(r^2\) means radius times radius

So to find the area of a circle, follow these steps:

1. Find the radius.
2. Square the radius.
3. Multiply by \(\pi\).

3. Important Reminder: Area and Circumference Are Not the Same

Students sometimes mix up area and circumference. Here is the difference:

• Area = space inside the circle
• Circumference = distance around the circle

If a question asks for how much surface is covered, you need area.

If a question asks for the distance around the edge, you need circumference.

4. Units for Area

Area is measured in square units.

• square centimeters = \(cm^2\)
• square meters = \(m^2\)
• square inches = \(in^2\)

The answer must be in square units because area measures a surface.

5. Worked Examples

Example 1: Radius is given

Find the area of a circle with radius \(4\) cm.

Use the formula:

$$A = \pi r^2$$

Substitute \(r = 4\):

$$A = \pi (4)^2$$

Square the radius:

$$A = \pi \cdot 16$$

Use \(\pi \approx 3.14\):

$$A \approx 3.14 \times 16 = 50.24$$

So the area is:

$$50.24\text{ cm}^2$$

Example 2: A larger radius

Find the area of a circle with radius \(7\) m.

Start with the formula:

$$A = \pi r^2$$

Substitute \(r = 7\):

$$A = \pi (7)^2$$

Square the radius:

$$A = \pi \cdot 49$$

Use \(\pi \approx 3.14\):

$$A \approx 3.14 \times 49 = 153.86$$

So the area is:

$$153.86\text{ m}^2$$

Example 3: Diameter is given instead of radius

Find the area of a circle with diameter \(12\) in.

First, find the radius:

$$r = \frac{d}{2} = \frac{12}{2} = 6$$

Now use the area formula:

$$A = \pi r^2$$

Substitute \(r = 6\):

$$A = \pi (6)^2$$

Square the radius:

$$A = \pi \cdot 36$$

Use \(\pi \approx 3.14\):

$$A \approx 3.14 \times 36 = 113.04$$

So the area is:

$$113.04\text{ in}^2$$

Example 4: Word problem

A circular garden has a radius of \(5\) feet. What is the area of the garden?

Use the formula:

$$A = \pi r^2$$

Substitute \(r = 5\):

$$A = \pi (5)^2$$

Square the radius:

$$A = \pi \cdot 25$$

Use \(\pi \approx 3.14\):

$$A \approx 3.14 \times 25 = 78.5$$

So the area of the garden is:

$$78.5\text{ ft}^2$$

6. Common Mistakes to Avoid

• Using the diameter instead of the radius
  Always check whether the number given is the radius or diameter.
• Forgetting to square the radius
  The formula is \(A = \pi r^2\), not \(A = \pi r\).
• Confusing area with circumference
  Area is inside the circle. Circumference is around it.
• Forgetting square units
  Area answers should be written in units like \(cm^2\), \(m^2\), or \(in^2\).

7. Quick Step-by-Step Method

1. Read the problem carefully.
2. Find the radius. If only the diameter is given, divide by 2.
3. Use the formula $$A = \pi r^2$$
4. Square the radius.
5. Multiply by \(3.14\) if needed.
6. Write the answer with square units.

8. Brief Summary

The area of a circle tells how much space is inside the circle. To find it, use the formula $$A = \pi r^2$$ where \(r\) is the radius.

If the diameter is given, divide it by 2 first to get the radius. Then square the radius, multiply by \(\pi\), and write the answer in square units.

Constant Perimeter with Changing Area

Sometimes two shapes can have the same perimeter but different areas. This is an important idea in measurement.

Perimeter is the distance around a shape. If you walked all the way around the edge, the total distance is the perimeter.

Area is the amount of space inside a shape. It tells how many square units cover the inside.

In this lesson, we will learn that when the perimeter stays the same, the area can still change. We will look at rectangles to understand this idea.

Big idea: A shape can keep the same border length, but the inside space can grow or shrink.

Let’s start by remembering how to find perimeter and area of a rectangle.

• Perimeter of a rectangle: add all 4 sides.
• Area of a rectangle: multiply length by width.

For a rectangle with length \, and width \, we can write:

$$P = l + w + l + w = 2l + 2w$$ $$A = l \times w$$

You do not need to memorize the letters. Just remember:

• Perimeter = distance around
• Area = space inside

Now let’s see what happens when the perimeter stays the same.

Suppose we have 12 units of border. That means the perimeter is always 12 units.

We can make different rectangles with perimeter 12:

• 1 by 5
• 2 by 4
• 3 by 3

Check the perimeter of each one:

• 1 by 5: \(1+5+1+5=12\)
• 2 by 4: \(2+4+2+4=12\)
• 3 by 3: \(3+3+3+3=12\)

All of these rectangles have the same perimeter: 12 units.

But now look at the area:

• 1 by 5: \(1 \times 5 = 5\) square units
• 2 by 4: \(2 \times 4 = 8\) square units
• 3 by 3: \(3 \times 3 = 9\) square units

The perimeter stayed the same, but the area changed.

This shows that same perimeter does not mean same area.

Let’s think about why this happens.

If one side gets very short and the other side gets very long, the rectangle becomes skinny. Skinny rectangles usually have less area.

If the side lengths are closer together, the rectangle looks more like a square. For a fixed perimeter, a square gives the greatest area.

So for rectangles with the same perimeter:

• More square-like shapes have more area.
• More skinny shapes have less area.

This is a very useful pattern.

Worked Example 1

A rectangle has perimeter 16 units. Find the area of a rectangle that is 1 unit by 7 units.

Step 1: Check the perimeter.

$$1+7+1+7=16$$

Yes, the perimeter is 16 units.

Step 2: Find the area.

$$1 \times 7 = 7$$

The area is 7 square units.

Worked Example 2

Another rectangle also has perimeter 16 units. Its side lengths are 2 units and 6 units. What is its area?

Step 1: Check the perimeter.

$$2+6+2+6=16$$

Step 2: Find the area.

$$2 \times 6 = 12$$

The area is 12 square units.

Compare Example 1 and Example 2:

• Both have perimeter 16 units.
• The 1 by 7 rectangle has area 7 square units.
• The 2 by 6 rectangle has area 12 square units.

Even though the border length stayed the same, the inside space got bigger.

Worked Example 3

Find the rectangle with perimeter 16 units that has the greatest area.

Let’s list the whole-number side lengths that work:

• 1 by 7
• 2 by 6
• 3 by 5
• 4 by 4

Now find each area:

• \(1 \times 7 = 7\)
• \(2 \times 6 = 12\)
• \(3 \times 5 = 15\)
• \(4 \times 4 = 16\)

The greatest area is 16 square units.

So the rectangle with the greatest area is 4 by 4, which is a square.

Worked Example 4

A farmer has 20 meters of fencing to make a rectangular garden. The fencing is the perimeter. Which garden gives more planting space: 1 m by 9 m, 2 m by 8 m, or 5 m by 5 m?

Step 1: Check that each garden has perimeter 20 meters.

• 1 by 9: \(1+9+1+9=20\)
• 2 by 8: \(2+8+2+8=20\)
• 5 by 5: \(5+5+5+5=20\)

Step 2: Find the area of each garden.

• 1 by 9: \(1 \times 9 = 9\) square meters
• 2 by 8: \(2 \times 8 = 16\) square meters
• 5 by 5: \(5 \times 5 = 25\) square meters

Step 3: Compare the areas.

The 5 m by 5 m garden gives the most planting space.

This is another example that shows a square gives the greatest area when the perimeter stays fixed.

How to think about these problems

1. Read the perimeter carefully.
2. Make sure the side lengths really add to that perimeter.
3. Find the area by multiplying the two side lengths.
4. Compare the areas.

Important pattern to remember

• Same perimeter can make different areas.
• Rectangles that are closer to a square have greater area.
• Rectangles that are long and skinny have smaller area.

Common mistake

Some students think that if two shapes have the same perimeter, they must also have the same area. That is not true.

For example:

• 2 by 6 has perimeter 16 and area 12
• 4 by 4 has perimeter 16 and area 16

Same perimeter. Different area.

Try it yourself

If the perimeter is 18 units, what whole-number rectangles can you make?

You can list them:

• 1 by 8
• 2 by 7
• 3 by 6
• 4 by 5

Now find the area of each:

• \(1 \times 8 = 8\)
• \(2 \times 7 = 14\)
• \(3 \times 6 = 18\)
• \(4 \times 5 = 20\)

The rectangle with the greatest area is 4 by 5.

Notice that 4 and 5 are close together. That rectangle is the least skinny.

Summary

Perimeter is the distance around a shape. Area is the space inside a shape.

When the perimeter stays the same, the area can change. For rectangles, shapes that are more like squares have larger areas, and long skinny rectangles have smaller areas.

So if you keep the border the same, you can still change how much space is inside.