Two-Dimensional Geometry

Relative Positional Vocabulary helps us tell where things are.

We use position words every day. We might say a bird is above a tree, or a toy is below a chair.

These words help us describe the location of objects clearly. In this lesson, we will learn and practice these words:

• above
• below
• beside
• next to
• in front of
• behind

Let’s learn what each word means.

Above means higher than something else.

Example: The sun is above the house.

Below means lower than something else.

Example: The rug is below the table.

Beside means at the side of something.

Example: The lamp is beside the bed.

Next to means almost the same as beside. It means right by the side of something.

Example: The cat is next to the box.

In front of means ahead of something.

Example: The child is in front of the slide.

Behind means at the back of something.

Example: The tree is behind the fence.

It helps to look carefully at two things and ask, “Where is one object compared to the other?”

Now let’s look at some easy examples.

Worked Example 1

A ball is on the floor. A shelf is higher up on the wall.

Question: Is the shelf above or below the ball?

Answer: The shelf is above the ball.

Why? The shelf is higher than the ball.

Worked Example 2

A dog is sitting by a tree.

Question: Is the dog beside the tree or behind the tree?

Answer: The dog is beside the tree.

Why? The dog is at the side of the tree, not at the back.

Worked Example 3

A boy is standing ahead of a bus stop sign.

Question: Is the boy in front of the sign or behind the sign?

Answer: The boy is in front of the sign.

Why? He is ahead of it.

Worked Example 4

A teddy bear is on a bed. A pillow is right by the teddy bear.

Question: Is the pillow next to the teddy bear or below the teddy bear?

Answer: The pillow is next to the teddy bear.

Why? It is right by the side of the teddy bear.

Let’s think about these word pairs:

• above and below are opposites.
• in front of and behind are opposites.
• beside and next to mean almost the same thing.

When you answer a position question, follow these steps:

1. Look at the two objects.
2. Choose one object to describe.
3. Compare its place to the other object.
4. Use the correct position word.

For example, if a bird is higher than a branch, we say the bird is above the branch.

If a shoe is at the side of a chair, we say the shoe is beside the chair.

If a toy car is at the back of a box, we say the toy car is behind the box.

You can also use these words when talking about shapes and pictures. In geometry, we often describe where a shape is placed.

Example: A square may be above a circle. A triangle may be next to a rectangle.

This helps us understand pictures and arrange shapes correctly.

Let’s practice with shape sentences:

• The circle is above the square.
• The triangle is below the rectangle.
• The oval is beside the hexagon.
• The star is in front of the box.
• The heart is behind the book.

Remember, position words tell us about location. They help us explain where something is.

Summary

• Above means higher than.
• Below means lower than.
• Beside means at the side of.
• Next to means right by the side of.
• In front of means ahead of.
• Behind means at the back of.

When you see objects or shapes, ask yourself: “Where is it?”

Then use the right positional word to describe it clearly.

You are learning to be great at telling where things are!

Identifying Basic 2D Shapes

Today we will learn about 2D shapes. A 2D shape is a flat shape. You can draw it on paper. It does not have thickness like a box or a ball.

We will learn to पहचान and name these shapes:

• circle
• triangle
• square
• rectangle
• hexagon
• rhombus

When we look at shapes, we can ask:

• How many sides does it have?
• How many corners does it have?
• Are the sides the same length or not?
• Does it have straight sides or a curved side?

Important shape words

• A side is a straight line on the edge of a shape.
• A corner is where two sides meet.
• A curve is round, not straight.

1. Circle

A circle is round. It has no straight sides and no corners. It is one curved line all the way around.

Think of a clock, a coin, or a round plate.

2. Triangle

A triangle has 3 sides and 3 corners.

If you count, you get:

$$3\text{ sides},\ 3\text{ corners}$$

Triangles can look different. Some are tall. Some are wide. But if a shape has 3 straight sides and 3 corners, it is a triangle.

3. Square

A square has 4 sides and 4 corners. All 4 sides are the same length.

A square looks like a box shape, but all its sides match.

4. Rectangle

A rectangle has 4 sides and 4 corners. It has 2 long sides and 2 short sides.

The opposite sides are the same length. That means the top and bottom match, and the left and right match.

5. Hexagon

A hexagon has 6 sides and 6 corners.

Count them carefully:

$$6\text{ sides},\ 6\text{ corners}$$

Some hexagons look like the shape on a honeycomb.

6. Rhombus

A rhombus has 4 sides and 4 corners. All 4 sides are the same length.

A rhombus may look like a square that is leaning or tilted. It is sometimes called a diamond shape.

How to tell shapes apart

Some shapes look a little alike. Let’s learn how to compare them.

• Circle vs. others: A circle has no corners. The other shapes here have corners.
• Triangle vs. quadrilaterals: A triangle has 3 sides. Squares, rectangles, and rhombuses have 4 sides.
• Square vs. rectangle: Both have 4 sides and 4 corners. A square has 4 equal sides. A rectangle has 2 long sides and 2 short sides.
• Square vs. rhombus: Both have 4 equal sides. A rhombus may look slanted or tilted.
• Hexagon: A hexagon is easy to spot because it has 6 sides.

Shape clue chart

• Circle: 0 sides, 0 corners, round
• Triangle: 3 sides, 3 corners
• Square: 4 equal sides, 4 corners
• Rectangle: 4 sides, 4 corners, 2 long and 2 short
• Hexagon: 6 sides, 6 corners
• Rhombus: 4 equal sides, 4 corners, tilted like a diamond

Worked Example 1

Question: I have a shape with no corners. It is round. What shape is it?

Think: A round shape with no corners is a circle.

Answer: circle

Worked Example 2

Question: A shape has 3 sides and 3 corners. What is it?

Think: Only one basic shape here has 3 sides.

Answer: triangle

Worked Example 3

Question: A shape has 4 corners. All 4 sides are the same length. It does not look like a box standing straight. It looks tilted. What shape is it?

Think: A shape with 4 equal sides could be a square or a rhombus. If it looks tilted like a diamond, it is a rhombus.

Answer: rhombus

Worked Example 4

Question: A shape has 6 sides. What shape is it?

Think: Count the sides: \(1,2,3,4,5,6\). A 6-sided shape is a hexagon.

Answer: hexagon

Let’s practice thinking

1. If a shape is round, it is a circle.
2. If a shape has 3 sides, it is a triangle.
3. If a shape has 4 equal sides and stands straight, it can be a square.
4. If a shape has 4 sides with 2 long and 2 short, it is a rectangle.
5. If a shape has 6 sides, it is a hexagon.
6. If a shape has 4 equal sides and looks like a diamond, it is a rhombus.

Look for shapes in real life

• A pizza can look like a circle.
• A slice of pizza can look like a triangle.
• A window can look like a square or rectangle.
• A honeycomb shape can look like a hexagon.
• A kite shape can look like a rhombus.

Tips for identifying shapes

• First, look at the shape carefully.
• Next, count the sides.
• Then, count the corners.
• Last, notice if the sides are all the same or not.

Summary

2D shapes are flat shapes. We can identify them by looking at their sides, corners, and whether they are round or straight.

• A circle is round with no corners.
• A triangle has 3 sides.
• A square has 4 equal sides.
• A rectangle has 2 long sides and 2 short sides.
• A hexagon has 6 sides.
• A rhombus has 4 equal sides and often looks like a diamond.

Now you can look at a flat shape, count its sides and corners, and name it!

Defining Attributes of 2D Shapes

Today we will learn about 2D shapes. A 2D shape is a flat shape. You can draw it on paper.

We will learn how to tell shapes apart by their defining attributes. That means the special parts of a shape that always stay the same.

For 1st grade, the most important defining attributes are:

• How many straight sides a shape has
• How many corners a shape has
• Whether the shape is closed, which means all the sides join up

These attributes help us name shapes correctly.

What is a side?

A side is a straight line that makes part of a shape.

What is a corner?

A corner is where two sides meet. Some people also call corners vertices.

What does closed mean?

A shape is closed when all the sides connect and there are no gaps.

If a drawing has a gap, it is not a closed shape.

Main Idea: A shape is named by its defining attributes, not by how big it is, what color it is, or which way it is turned.

A triangle is still a triangle if it is small, big, red, blue, tilted, or turned.

Let’s learn some common 2D shapes.

• Triangle: 3 straight sides and 3 corners
• Rectangle: 4 straight sides and 4 corners
• Square: 4 straight sides and 4 corners
• Circle: no straight sides and no corners

Let’s count the attributes for each one:

• Triangle: $$3\text{ sides},\ 3\text{ corners}$$
• Rectangle: $$4\text{ sides},\ 4\text{ corners}$$
• Square: $$4\text{ sides},\ 4\text{ corners}$$
• Circle: $$0\text{ straight sides},\ 0\text{ corners}$$

Important: A square and a rectangle both have 4 straight sides and 4 corners. They are both closed shapes.

What does not matter?

• Color
• Size
• Where the shape is on the page
• If the shape is turned or flipped

What does matter?

• Number of straight sides
• Number of corners
• If the shape is closed

Worked Example 1

A shape has 3 straight sides. It has 3 corners. All the sides connect.

What shape is it?

Step 1: Count the sides. There are 3.

Step 2: Count the corners. There are 3.

Step 3: Check if it is closed. Yes, it is.

Answer: It is a triangle.

Worked Example 2

A shape has 4 straight sides and 4 corners. It is closed.

What could the shape be?

Step 1: Count the sides. There are 4.

Step 2: Count the corners. There are 4.

Step 3: Check if it is closed. Yes.

Answer: It could be a square or a rectangle.

Worked Example 3

A drawing has 4 lines, but one line does not touch the next line. There is a gap.

Is it a shape we can name?

Step 1: Look for a gap.

Step 2: See that the lines do not all join.

Answer: No. It is not a closed shape.

Worked Example 4

A shape is round. It has no straight sides. It has no corners.

What shape is it?

Step 1: Count straight sides. There are 0.

Step 2: Count corners. There are 0.

Answer: It is a circle.

Let’s compare shapes.

1. Triangle and rectangle
   • A triangle has 3 sides and 3 corners.
   • A rectangle has 4 sides and 4 corners.
   • They are different because their defining attributes are different.
2. Square and rectangle
   • Both have 4 straight sides.
   • Both have 4 corners.
   • Both are closed shapes.
3. Circle and triangle
   • A circle has 0 straight sides and 0 corners.
   • A triangle has 3 straight sides and 3 corners.
   • They are very different shapes.

How to classify a 2D shape

When you see a shape, ask yourself these questions:

1. Is it closed?
2. How many straight sides does it have?
3. How many corners does it have?
4. What shape name matches those attributes?

Try this thinking:

• If it has 3 straight sides and 3 corners, it is a triangle.
• If it has 4 straight sides and 4 corners, it may be a square or rectangle.
• If it is round with no straight sides and no corners, it is a circle.
• If it has a gap, it is not closed.

Building shapes

You can also make shapes using sticks, straws, or drawing lines.

• To make a triangle, use 3 straight lines and connect them.
• To make a rectangle, use 4 straight lines and connect them.
• To make a square, use 4 straight lines and connect them.

If the lines do not connect all the way around, the shape is not closed.

Be careful!

• A shape does not stop being a triangle just because it is turned.
• A big circle and a small circle are both circles.
• A blue square and a red square are both squares.

The defining attributes stay the same even when the shape looks different in size, color, or direction.

Summary

2D shapes are flat shapes. We can sort and name them by their defining attributes.

The most important defining attributes are the number of straight sides, the number of corners, and whether the shape is closed.

Remember:

• Triangle: 3 sides, 3 corners
• Square: 4 sides, 4 corners
• Rectangle: 4 sides, 4 corners
• Circle: 0 straight sides, 0 corners

When you look at a shape, count carefully and check if it is closed. That will help you name the shape correctly.

Non-Defining Attributes

Today we will learn about non-defining attributes of shapes.

That sounds like a big idea, but it is simple. A shape can look different in some ways and still be the same kind of shape.

Some things about a shape do not change its name. These are called non-defining attributes.

For example, a triangle is still a triangle if it is big or small, red or blue, plain or striped, standing up or turned sideways.

What can change and still keep the same shape name?

• Size — big or small
• Color — red, blue, green, or any color
• Pattern — plain, striped, dotted, or checked
• Orientation — turned, flipped, or tilted

These things do not tell us the shape name.

What does tell us the shape name?

We look at the shape itself.

• A triangle has 3 straight sides.
• A square has 4 equal sides.
• A rectangle has 4 sides.
• A circle is round.

These are the important parts that help us know the shape. The color or size does not change that.

Think about it like this:

If you wear a red shirt one day and a blue shirt the next day, you are still you. In the same way, if a square is blue or yellow, it is still a square.

Main Idea

A shape keeps its name when only the non-defining attributes change.

We can say:

$$\text{same shape} = \text{same name}$$

even if the size, color, pattern, or direction changes.

Let’s learn each non-defining attribute.

1. Size

A shape can be big or small.

A big triangle and a small triangle are both triangles.

The size changed, but the shape name did not change.

2. Color

A shape can be any color.

A red circle and a green circle are both circles.

The color changed, but the shape name did not change.

3. Pattern

A shape can have stripes, dots, or no pattern at all.

A striped square and a plain square are both squares.

The pattern changed, but the shape name did not change.

4. Orientation

A shape can be turned.

Sometimes children think a shape has a new name when it is turned. But turning a shape does not change its name.

A square turned a little is still a square. A triangle turned upside down is still a triangle.

Worked Example 1

You see a small blue triangle and a big yellow triangle.

Step 1: Look at the shape, not the color.

Step 2: Look at the sides. Each shape has 3 straight sides.

Step 3: Decide the name.

Both shapes are triangles.

Why? Size and color are non-defining attributes.

Worked Example 2

You see a square standing flat and another square turned like a diamond.

Step 1: Ask, “Was the shape only turned?”

Step 2: Check the shape. It still has 4 equal sides.

Step 3: Name the shape.

It is still a square.

Why? Turning a shape does not change its name.

Worked Example 3

You see two circles. One is red with dots. One is green and plain.

Step 1: Ignore the color and pattern.

Step 2: Look at the shape. Both are round.

Step 3: Name the shape.

Both are circles.

Why? Color and pattern are non-defining attributes.

Worked Example 4

You see a small rectangle and a large rectangle turned sideways.

Step 1: Check if only the size and direction changed.

Step 2: Look at the shape. Both have 4 sides and are rectangles.

Step 3: Name the shapes.

Both are rectangles.

Why? Size and orientation do not change the shape name.

How to identify a shape correctly

1. Look at the shape.
2. Do not worry about the color.
3. Do not worry about the size.
4. Do not worry about stripes or dots.
5. Do not worry if the shape is turned.
6. Look at the sides and corners, or see if it is round.
7. Name the shape.

Let’s compare.

These can change:

• big or small
• red or blue
• striped or plain
• upright or turned

These help tell the shape name:

• how many sides it has
• whether the sides are straight
• whether it is round
• for a square, that the sides are equal

Easy shape reminders

• Triangle → 3 sides
• Square → 4 equal sides
• Rectangle → 4 sides
• Circle → round

Try thinking about these:

• If a circle is purple, is it still a circle? Yes.
• If a triangle is tiny, is it still a triangle? Yes.
• If a square is turned, is it still a square? Yes.
• If a rectangle has stripes, is it still a rectangle? Yes.

Summary

Non-defining attributes are things that do not change a shape’s name.

These include size, color, pattern, and orientation.

A shape is named by what kind of shape it is, not by how big it is, what color it is, what pattern it has, or how it is turned.

So remember:

Turn it, color it, shrink it, or make it bigger — it can still be the same shape.

Composing 2D Shapes means putting flat shapes together to make a new shape. A flat shape is a shape that lies flat on a page, like a square, triangle, rectangle, or circle.

In this lesson, we will learn how to look at small shapes and see how they can join to make one bigger shape. This is like building with pattern blocks or puzzle pieces.

Some 2D shapes we know:

• Triangle — has 3 straight sides.
• Square — has 4 equal sides.
• Rectangle — has 4 sides. It is longer in one direction and shorter in the other.
• Circle — is round and has no straight sides.

When we compose shapes, we join two or more shapes together. The small shapes do not disappear. They are still there, but together they make a new, larger shape.

For example, two triangles can be put together to make a square or a rectangle. Two squares can be put side by side to make a rectangle.

How to compose shapes

1. Look at the small shapes.
2. Notice their sides and corners.
3. Think about how the shapes can touch.
4. Put them together to make a bigger shape.
5. Check the new shape: What shape did you make?

Important idea: Shapes can be turned or flipped and still be the same shape. A triangle is still a triangle if it points up, down, or sideways.

If two shapes fit together with no gaps, they can make a new shape. Sometimes the new shape is a shape you already know, like a square or rectangle. Sometimes it is just a bigger new shape made from smaller parts.

Worked Example 1

You have 2 small squares. Put them side by side.

Each square has 4 sides. When the squares touch along one side, they make a longer shape.

The new larger shape is a rectangle.

We can think of it like this:

\(1\) square + \(1\) square = \(1\) rectangle

Worked Example 2

You have 2 same-size triangles. Put them together along one side.

If the triangles are joined the right way, they can make a square.

So, 2 triangles can make 1 square.

This shows us that bigger shapes can be made from smaller shapes.

Worked Example 3

You have 1 square and 2 triangles.

Put the square in the middle. Put one triangle on one side and the other triangle on the other side.

Now you have made a larger shape. It may look like a house.

The new shape is made from 3 smaller shapes:

• 1 square
• 2 triangles

This is called a composite shape because it is made by putting shapes together.

Worked Example 4

You have 4 small triangles.

If you join them carefully, they can make 1 larger triangle.

That means small shapes and large shapes can have the same name. A small triangle and a large triangle are both triangles.

What to look for when shapes are put together

• Sides — Do the sides match up?
• Corners — Do the corners meet neatly?
• Gaps — Are there empty spaces? If yes, the shape is not fully put together.
• New shape — Does the larger shape look like a square, rectangle, triangle, or something else?

Let’s think together.

If you put 2 circles next to each other, do they make a rectangle? No. Circles are round, so they do not make straight sides like a rectangle.

If you put 2 squares together, can they make a rectangle? Yes. Their straight sides match up well.

If you put 2 triangles together, can they make a bigger shape? Yes. They can make a square, rectangle, or a larger triangle, depending on how they are placed.

Try these ideas in your head:

• 2 squares can make 1 rectangle.
• 2 triangles can make 1 square.
• 4 triangles can make 1 larger triangle.
• A square and triangles can make a house-like shape.

Tips for composing shapes

• Move the shape around if it does not fit at first.
• Turn the shape to see if it matches better.
• Look for straight sides that can touch.
• Check that there are no holes or gaps.

Why this matters

Composing shapes helps us see how small parts make a whole. It helps us solve puzzles, build pictures, and understand how shapes work together.

You can practice with paper shapes, blocks, or drawings. Try making a rectangle from 2 squares. Try making a square from triangles. Try making your own picture from shapes.

Summary

Composing 2D shapes means putting two or more flat shapes together to make a new, larger shape. We can use shapes like squares, rectangles, triangles, and circles. We look at sides, corners, and gaps to see if shapes fit together. When shapes are joined, they make a composite shape.

Decomposing 2D Shapes means taking one flat shape and breaking it into smaller flat shapes.

A 2D shape is a flat shape. Some 2D shapes you may know are a square, rectangle, triangle, and circle.

When we decompose a shape, we look for smaller shapes hiding inside a bigger shape. Then we can draw lines or imagine cutting the shape apart.

This helps us see shapes in new ways. A big shape can be made from 2 smaller shapes, 3 smaller shapes, or even more.

Main Idea: One larger shape can be split into smaller shapes.

• A rectangle can be split into 2 smaller rectangles.
• A square can be split into 2 triangles.
• A bigger shape can be made from squares, rectangles, and triangles.

We can ask ourselves:

• What shape do I see first?
• Can I draw a line to make smaller shapes?
• What smaller shapes do I see now?

Let’s learn how.

1. Look at the whole shape.
2. Find parts that look like smaller shapes.
3. Draw a line, or imagine a cut.
4. Name the smaller shapes you made.

Example 1: A rectangle split into 2 smaller rectangles

Imagine one big rectangle.

If we draw one line straight down the middle, we make 2 parts.

Each part is a rectangle.

So, 1 rectangle can be decomposed into:

$$2 \text{ rectangles}$$

Worked Example 1

Shape: one large rectangle

Step 1: Look at the rectangle.

Step 2: Draw one line from top to bottom in the middle.

Step 3: Count the new shapes.

Answer: The large rectangle is decomposed into 2 rectangles.

Example 2: A square split into 2 triangles

Imagine a square.

If we draw a line from one corner to the opposite corner, that line is called a diagonal. It splits the square into 2 smaller shapes.

Those 2 smaller shapes are triangles.

So, 1 square can be decomposed into:

$$2 \text{ triangles}$$

Worked Example 2

Shape: one square

Step 1: Look at the square.

Step 2: Draw one slanted line from one corner to the opposite corner.

Step 3: Name the new shapes.

Answer: The square is decomposed into 2 triangles.

Example 3: A larger rectangle split into 4 squares

Now imagine a larger rectangle made like a window.

Draw 1 line up and down and 1 line across.

The rectangle is now split into 4 smaller parts.

If each part has equal sides, each small part is a square.

So, the larger shape can be decomposed into:

$$4 \text{ squares}$$

Worked Example 3

Shape: one large rectangle or square

Step 1: Draw one vertical line.

Step 2: Draw one horizontal line.

Step 3: Count the smaller parts.

Step 4: Name them.

Answer: The shape is decomposed into 4 smaller squares.

Example 4: A house shape

Sometimes a shape is made from different smaller shapes.

Think of a simple house picture. The bottom part looks like a rectangle. The roof looks like a triangle.

So the house shape can be decomposed into:

$$1 \text{ rectangle} + 1 \text{ triangle}$$

Worked Example 4

Shape: house shape

Step 1: Look at the bottom. It is a rectangle.

Step 2: Look at the top. It is a triangle.

Step 3: Name both smaller shapes.

Answer: The house shape is decomposed into 1 rectangle and 1 triangle.

Tips for decomposing shapes

• Go slowly and look carefully.
• Use your finger to trace the shape.
• Ask, “Can I split this shape into smaller parts?”
• Look for straight lines you can draw.
• Count the smaller shapes after you split the big shape.

Let’s practice thinking.

If you see a square, can you split it into 2 triangles? Yes.

If you see a rectangle, can you split it into 2 rectangles? Yes.

If you see a big shape that looks like a house, can you find a rectangle and a triangle? Yes.

Why this matters

When you decompose shapes, you become a shape detective. You learn that big shapes can be made from small shapes.

This helps you when you draw, build, and solve math problems with shapes.

Summary

Decomposing 2D shapes means breaking one flat shape into smaller flat shapes.

You can decompose shapes by drawing lines or imagining cuts.

A rectangle can become 2 rectangles, a square can become 2 triangles, and a larger shape can be made from different smaller shapes like a rectangle and a triangle.

Look carefully, split the shape, and name the smaller shapes you see.