3D Geometry: Surface Area and Volume

Identifying Polyhedra, Faces, Edges, and Vertices

When we move from flat shapes to solid shapes, we begin studying 3D geometry. A 3D shape has length, width, and height. Some 3D shapes are made only of flat surfaces. These special solids are called polyhedra.

In this lesson, you will learn how to recognize a polyhedron and how to identify its faces, edges, and vertices. You will also learn how these parts help us name and classify solids such as prisms and pyramids.

What is a polyhedron?

A polyhedron is a 3D solid made up of flat polygon faces. A polygon is a flat shape with straight sides, such as a triangle, rectangle, square, or pentagon.

For a solid to be a polyhedron:

• All of its surfaces must be flat.
• Its surfaces must be polygons.
• Its sides must be straight, not curved.

Examples of polyhedra:

• Cube
• Rectangular prism
• Triangular prism
• Square pyramid

Not polyhedra:

• Sphere
• Cylinder
• Cone

These are not polyhedra because they have curved surfaces.

The parts of a polyhedron

Every polyhedron has important parts that help us describe it.

• Face: A flat surface of a solid.
• Edge: A line segment where two faces meet.
• Vertex: A point where edges meet. The plural of vertex is vertices.

Think of a cardboard box:

• Each flat side is a face.
• The lines where the sides join are edges.
• The corners are vertices.

How to count faces, edges, and vertices

When counting parts of a solid, go slowly and use a system so you do not count the same part twice.

1. Count the faces first by finding each flat surface.
2. Count the edges by tracing each line where two faces meet.
3. Count the vertices by counting each corner point.

Sometimes it helps to imagine turning the solid in your hands so you can see the hidden parts on the back or bottom.

Prisms

A prism is a polyhedron with:

• Two matching, parallel faces called bases
• Side faces that connect the bases

The prism is named by the shape of its bases.

• A triangular prism has triangular bases.
• A rectangular prism has rectangular bases.
• A pentagonal prism has pentagonal bases.

In a prism, the two bases are the same shape and size.

Pyramids

A pyramid is a polyhedron with:

• One base
• Triangular faces that meet at one top point

That top point is called an apex. The pyramid is named by the shape of its base.

• A triangular pyramid has a triangular base.
• A square pyramid has a square base.
• A pentagonal pyramid has a pentagonal base.

Common polyhedra and their parts

Here are some important solids to know:

• Cube: 6 faces, 12 edges, 8 vertices
• Rectangular prism: 6 faces, 12 edges, 8 vertices
• Triangular prism: 5 faces, 9 edges, 6 vertices
• Square pyramid: 5 faces, 8 edges, 5 vertices
• Triangular pyramid: 4 faces, 6 edges, 4 vertices

Worked Example 1: Cube

Question: How many faces, edges, and vertices does a cube have?

Step 1: Count the faces. A cube has a top, bottom, front, back, left side, and right side. That is 6 faces.

Step 2: Count the edges. A cube has 4 edges on the top, 4 edges on the bottom, and 4 vertical edges connecting them.

$$4+4+4=12$$

So, a cube has 12 edges.

Step 3: Count the vertices. A cube has 4 corners on top and 4 corners on bottom.

$$4+4=8$$

So, a cube has 8 vertices.

Answer: A cube has 6 faces, 12 edges, and 8 vertices.

Worked Example 2: Triangular Prism

Question: How many faces, edges, and vertices does a triangular prism have?

Step 1: Think about the bases. A triangular prism has 2 triangular bases.

Step 2: Count the side faces. Each side of the triangle connects to the matching side on the other triangle. A triangle has 3 sides, so there are 3 rectangular side faces.

Total faces:

$$2+3=5$$

So, the prism has 5 faces.

Step 3: Count the edges.

• 3 edges on the front triangle
• 3 edges on the back triangle
• 3 edges connecting the triangles

$$3+3+3=9$$

So, the triangular prism has 9 edges.

Step 4: Count the vertices.

• 3 vertices on one triangle
• 3 vertices on the other triangle

$$3+3=6$$

So, the triangular prism has 6 vertices.

Answer: A triangular prism has 5 faces, 9 edges, and 6 vertices.

Worked Example 3: Square Pyramid

Question: How many faces, edges, and vertices does a square pyramid have?

Step 1: Count the faces.

• 1 square base
• 4 triangular side faces

$$1+4=5$$

So, the square pyramid has 5 faces.

Step 2: Count the edges.

• 4 edges around the square base
• 4 edges from the base corners to the apex

$$4+4=8$$

So, the square pyramid has 8 edges.

Step 3: Count the vertices.

• 4 vertices on the square base
• 1 apex

$$4+1=5$$

So, the square pyramid has 5 vertices.

Answer: A square pyramid has 5 faces, 8 edges, and 5 vertices.

Worked Example 4: Is it a polyhedron?

Question: Which of these is a polyhedron: a cylinder or a rectangular prism?

Step 1: Check the surfaces.

• A cylinder has a curved surface.
• A rectangular prism has only flat rectangular faces.

Step 2: Use the definition. A polyhedron must have only flat polygon faces.

Answer: The rectangular prism is a polyhedron. The cylinder is not.

Helpful patterns

You may notice patterns when comparing solids.

• Prisms have two matching bases.
• Pyramids have one base and all side faces meet at one point.
• If a solid has a curved surface, it is not a polyhedron.

These patterns can help you identify a shape even before you count its parts.

Tips for avoiding mistakes

• Do not count curved surfaces as faces of a polyhedron.
• Do not forget hidden edges or back vertices.
• Remember that vertices are corners, not sides.
• Remember that edges are where two faces meet.
• Look at the base to help name a prism or pyramid.

Quick check questions

1. Is a cone a polyhedron? No, because it has a curved surface.
2. How many vertices does a rectangular prism have? 8.
3. What shape are the bases of a triangular prism? Triangles.
4. How many faces does a triangular pyramid have? 4.

Summary

A polyhedron is a 3D solid made only of flat polygon faces. Its parts are faces (flat surfaces), edges (where faces meet), and vertices (corners).

To identify and classify polyhedra, look at the shape of the base and count the faces, edges, and vertices carefully. Prisms have two matching bases, while pyramids have one base and triangular sides that meet at a point.

Lesson: Nets of Three-Dimensional Figures

In 3D geometry, a net is a flat, 2D pattern that can be folded to make a 3D solid. You can think of a net as an unfolded box or solid.

Learning about nets helps you picture shapes in space. It also helps when you study surface area, because a net shows all the outside faces of a solid at once.

In this lesson, you will learn how to recognize nets, match nets to solids, and decide whether a flat pattern can really fold into a 3D figure.

1. What is a net?

A three-dimensional figure is a solid shape, such as a cube, rectangular prism, triangular prism, or pyramid. These solids are made of faces, which are flat surfaces.

A net shows all of those faces laid out flat. When you fold along the edges, the faces meet to form the solid.

• A cube has 6 square faces, so its net must have 6 squares.
• A rectangular prism has 6 rectangular faces, so its net must have 6 rectangles (some may be the same size).
• A triangular prism has 2 triangular faces and 3 rectangular faces.
• A square pyramid has 1 square base and 4 triangular faces.

2. Important idea: faces must match the solid

To decide whether a net matches a solid, first count the faces and check their shapes.

For example, a cube must have exactly 6 squares. If a pattern has 5 squares, it cannot be a cube. If it has 6 shapes but one of them is a triangle, it also cannot be a cube.

This is the first check: Do the number and shapes of the faces match the solid?

3. Important idea: the faces must fold without overlapping

Even if a pattern has the correct number of faces, it may still not be a net. When folded, the faces must meet properly to close the solid.

If two faces try to cover the same space, or if there is a gap, then the pattern is not a correct net.

So the second check is: Can the faces fold up and meet correctly?

4. Nets of common solids

Cube

A cube has 6 congruent square faces. Many different nets can fold into a cube, but every correct cube net uses 6 connected squares.

Rectangular prism

A rectangular prism also has 6 faces, but the faces may have different side lengths. Opposite faces are the same size. A net must show all 6 rectangles arranged so they can fold into a box shape.

Triangular prism

A triangular prism has 5 faces total:

• 2 triangles
• 3 rectangles

A correct net must include those exact faces.

Square pyramid

A square pyramid has 5 faces total:

• 1 square
• 4 triangles

The triangles fold up around the square base and meet at one top point.

5. How to identify a net

When you look at a flat pattern, use these steps:

1. Count the faces.
2. Check the shapes of the faces.
3. Picture the folding. Ask: Which face is the base? Which faces fold up around it?
4. Check for overlap or gaps. Do the faces close the solid exactly?

6. Worked Examples

Example 1: Is this a net for a cube?

A pattern has 6 equal squares connected edge-to-edge.

Step 1: A cube needs 6 square faces.

This pattern has 6 squares, so it passes the first check.

Step 2: Imagine folding the side squares up and the last square over the top.

If the squares meet without overlapping, then the pattern is a net of a cube.

Answer: Yes, it could be a cube net, as long as the arrangement folds correctly.

Example 2: Which solid matches this net?

A net has 1 square and 4 triangles.

Step 1: Count the faces.

There are 5 faces total.

Step 2: Match the face shapes to a solid.

• 1 square base
• 4 triangular sides

This matches a square pyramid.

Answer: The net folds into a square pyramid.

Example 3: Is this a net for a triangular prism?

A pattern has 2 triangles and 3 rectangles.

Step 1: A triangular prism has:

• 2 triangular faces
• 3 rectangular faces

The pattern has the correct number and type of faces.

Step 2: Imagine the 3 rectangles wrapping around the sides, while the 2 triangles close the ends.

If the rectangles form the side surface and the triangles fit on opposite ends, then it is a correct net.

Answer: Yes, this can be a net for a triangular prism.

Example 4: Why is this not a cube net?

A pattern has 6 squares, but when folded, two squares land on top of each other and one side is left open.

Step 1: The number of faces is correct.

Step 2: But the folding does not work.

• Two faces overlap
• The solid does not close properly

Answer: It is not a cube net, because a correct net must fold into the solid without overlap or gaps.

7. How nets connect to surface area

A net is useful because it shows every outside face of a solid. This makes it easier to find surface area.

For example, if a cube has side length \(4\) units, each square face has area

$$4 \times 4 = 16$$

Since a cube has 6 faces, the total surface area is

$$6 \times 16 = 96$$

The net helps you see all 6 faces at once.

8. Tips for success

• Count carefully. Make sure the net has the right number of faces.
• Check face shapes. A prism and a pyramid have different kinds of faces.
• Visualize folding. Try to imagine the faces standing up.
• Watch for overlap. A correct net must fold neatly.
• Use the base. Find the base first, then see what folds around it.

9. Quick review of common solids and their faces

• Cube: 6 squares
• Rectangular prism: 6 rectangles
• Triangular prism: 2 triangles and 3 rectangles
• Square pyramid: 1 square and 4 triangles

Summary

A net is a flat pattern that folds to make a 3D solid. To recognize a net, check the number of faces, the shapes of the faces, and whether the pattern can fold without gaps or overlap.

Nets are important because they help you understand solids and find surface area. When you can picture a solid unfolding and folding back up, you are building strong 3D geometry skills.

Surface Area of Prisms and Pyramids

When we look at a 3D shape, it has outside surfaces called faces. The surface area of a solid figure is the total area of all its faces.

You can think of surface area like the amount of paper needed to wrap a box or cover a solid shape. To find it, we add the areas of all the flat faces on the outside.

In this lesson, we will learn how to find the surface area of prisms and pyramids by using their nets.

1. What is a prism?

A prism is a 3D shape with:

• two matching faces called bases, and
• side faces that are rectangles.

Examples of prisms are rectangular prisms and triangular prisms.

2. What is a pyramid?

A pyramid is a 3D shape with:

• one base, and
• triangular faces that meet at one point at the top.

An example is a square pyramid, which has a square base and 4 triangular side faces.

3. What is a net?

A net is a flat pattern that shows all the faces of a 3D shape opened up. Nets are very helpful because they let us see every face clearly.

To find surface area from a net, follow these steps:

1. Identify every face.
2. Find the area of each face.
3. Add all the face areas together.

4. Area formulas you need

To find surface area, you need to know how to find the area of rectangles, squares, and triangles.

• Rectangle: \(A = l \times w\)
• Square: \(A = s \times s\)
• Triangle: \(A = \frac{1}{2}bh\)

In a triangle, \(b\) is the base and \(h\) is the height.

5. Surface area of a prism

A prism has two matching bases and some rectangular side faces. The total surface area is:

This means we add:

• the area of the first base,
• the area of the second base, and
• the areas of all the side rectangles.

Worked Example 1: Rectangular Prism

Find the surface area of a rectangular prism with length 6 cm, width 4 cm, and height 3 cm.

A rectangular prism has 6 faces:

• Top and bottom: each is \(6 \times 4\)
• Front and back: each is \(6 \times 3\)
• Left and right: each is \(4 \times 3\)

Find each pair of faces:

Add them:

The surface area is 108 square centimeters, or \(108\text{ cm}^2\).

Worked Example 2: Triangular Prism

A triangular prism has:

• 2 triangular bases, each with base 8 cm and height 5 cm
• 3 rectangular side faces with areas 24 cm², 40 cm², and 30 cm²

First find the area of one triangular base:

There are 2 triangles:

Now add the rectangular side faces:

Add everything together:

The surface area is 134 square centimeters, or \(134\text{ cm}^2\).

6. Surface area of a pyramid

A pyramid has one base and triangular side faces. To find its surface area, add:

• the area of the base, and
• the areas of all the triangles.

Worked Example 3: Square Pyramid

A square pyramid has:

• a square base with side length 10 cm
• 4 triangular faces, each with base 10 cm and height 6 cm

First find the area of the square base:

Now find the area of one triangle:

There are 4 triangles:

Add the base and the triangles:

The surface area is 220 square centimeters, or \(220\text{ cm}^2\).

Worked Example 4: Rectangular Prism from a Net

A net shows a rectangular prism with these faces:

• 2 rectangles of \(7 \text{ cm} \times 2 \text{ cm}\)
• 2 rectangles of \(7 \text{ cm} \times 5 \text{ cm}\)
• 2 rectangles of \(2 \text{ cm} \times 5 \text{ cm}\)

Find the area of each pair:

Add the areas:

The surface area is 118 square centimeters, or \(118\text{ cm}^2\).

7. Tips for solving surface area problems

• Use the net if one is given. It helps you see every face.
• Check that you counted all faces. Missing one face gives the wrong answer.
• Use the correct area formula for each face.
• Write square units in your answer, like \(\text{cm}^2\), \(\text{m}^2\), or \(\text{in}^2\).
• Add carefully. Surface area is the total of all outside faces.

8. Common mistakes to avoid

• Confusing surface area with volume. Surface area covers the outside; volume measures space inside.
• Forgetting one or more faces.
• Using \(l \times w\) for a triangle instead of \(\frac{1}{2}bh\).
• Forgetting to multiply when there are matching faces.
• Leaving off the square units.

9. Quick check

Ask yourself these questions when solving:

• What shape is each face?
• How many of each face are there?
• Did I find every area correctly?
• Did I add all the face areas?
• Did I write my answer in square units?

Summary

The surface area of a prism or pyramid is the total area of all its outside faces. A net helps us see those faces clearly.

For a prism, add the areas of the two bases and the rectangular side faces. For a pyramid, add the area of the base and the areas of the triangular faces.

If you work carefully, find each face area, and add them all together, you can find the surface area of many 3D shapes.

Volume of Rectangular Prisms

When we talk about volume, we mean how much space is inside a solid figure. A rectangular prism is a 3D shape like a box, a brick, or a cereal container. It has a length, width, and height.

To find the volume of a rectangular prism, we imagine filling it with tiny cubes called unit cubes. Each unit cube is 1 unit long, 1 unit wide, and 1 unit tall, so it has a volume of 1 cubic unit.

If a prism can be packed with 24 unit cubes, then its volume is 24 cubic units. Volume tells us the prism’s cubic capacity, or how much it can hold.

Important idea: Volume is measured in cubic units, such as cubic centimeters, cubic inches, or cubic feet. We write these as \(cm^3\), \(in^3\), or \(ft^3\).

The Formula

For a rectangular prism, the volume is found by multiplying the three dimensions:

Here:

• \(l\) = length
• \(w\) = width
• \(h\) = height

You can also think of volume in two steps:

1. Find the area of the base: \(l \times w\)
2. Multiply by the height: base area \(\times h\)

So another way to write the formula is:

where \(B\) means the area of the base.

Why the Formula Works

Imagine the bottom layer of a rectangular prism. If the base is 4 units long and 3 units wide, then one layer has:

unit cubes.

If the prism is 5 units high, then there are 5 layers of 12 cubes each:

So the prism has a volume of 60 cubic units.

This is why multiplying length, width, and height gives the total number of unit cubes that fit inside.

Steps for Finding Volume

1. Identify the length, width, and height.
2. Multiply the three numbers.
3. Write the answer in cubic units.

Worked Example 1

Find the volume of a rectangular prism with length 6 cm, width 4 cm, and height 3 cm.

Use the formula:

The volume is 72 cubic centimeters.

Answer: \(72\,cm^3\)

Worked Example 2

A storage box is 8 in long, 5 in wide, and 2 in high. What is its volume?

The volume is 80 cubic inches.

Answer: \(80\,in^3\)

Worked Example 3

A prism has a base that is 7 ft by 3 ft, and it is 4 ft tall. Find the volume.

First find the area of the base:

Now multiply by the height:

The volume is 84 cubic feet.

Answer: \(84\,ft^3\)

Worked Example 4: Fractional Edge Lengths

Rectangular prisms can also have side lengths that are fractions. The process is the same: multiply the length, width, and height.

Find the volume of a rectangular prism with length \(\frac{1}{2}\) m, width \(3\) m, and height \(2\) m.

Multiply step by step:

The volume is 3 cubic meters.

Answer: \(3\,m^3\)

You can also picture this prism as being filled with smaller fractional unit cubes. Even when the edges are fractions, volume still counts how much 3D space is inside.

Another Fraction Example

Find the volume of a rectangular prism with side lengths \(\frac{3}{4}\) in, \(2\) in, and \(4\) in.

The volume is 6 cubic inches.

Answer: \(6\,in^3\)

Common Mistakes to Avoid

• Forgetting one dimension: You must multiply length, width, and height.
• Using square units instead of cubic units: Volume is always in cubic units, not square units.
• Mixing up surface area and volume: Surface area measures the outside. Volume measures the space inside.
• Not labeling the answer: Always include units like \(cm^3\) or \(ft^3\).

Helpful Check

Ask yourself: Does my answer make sense for the size of the box? If the prism is fairly large, the volume should not be tiny. Also check that you multiplied all three dimensions.

Practice Thinking

• If a prism is 2 units by 2 units by 2 units, then \(2 \times 2 \times 2 = 8\). So it holds 8 unit cubes.
• If one dimension doubles, the volume also doubles, as long as the other two dimensions stay the same.
• If a side length is a fraction, the volume can still be a whole number or a fraction, depending on the dimensions.

Summary

Volume tells how much space is inside a rectangular prism. You can find it by counting unit cubes or by using the formula \(V = l \times w \times h\). Always remember to write the answer in cubic units.

Cross-Sections of Three-Dimensional Figures

When we cut a three-dimensional, or 3D, solid, the flat shape we see on the cut surface is called a cross-section.

A cross-section is always a 2D shape. That means it is flat, like a circle, rectangle, triangle, or oval.

Learning about cross-sections helps us understand how 3D shapes are built. It also helps us picture what happens when a solid is sliced in different ways.

Think about real life. If you slice a loaf of bread, each piece shows a cross-section. If you cut a cucumber, the cut face is also a cross-section. The shape you see depends on how you cut.

Main idea: The cross-section depends on two things:

• the 3D figure being sliced
• the direction of the slice

Let’s look at common 3D figures and the cross-sections they can make.

1. Cube and Rectangular Prism

A cube has square faces. A rectangular prism has rectangular faces. If you slice these solids, you can get different flat shapes.

• If you slice straight across, parallel to the top or bottom, you get a square or rectangle.
• If you slice straight up and down, parallel to a side face, you also get a rectangle.
• If you slice at a slant, you may get other shapes, but in 6th Grade we usually focus on simple cross-sections like squares and rectangles.

For a cube, a straight slice parallel to a face gives a square.

For a rectangular prism, a straight slice parallel to a face gives a rectangle.

2. Cylinder

A cylinder has two circular bases and one curved surface.

• If you slice it parallel to the circular base, the cross-section is a circle.
• If you slice it straight up and down through the cylinder, the cross-section is a rectangle.
• If you slice it at a slant, the cross-section can look like an oval.

So one 3D shape can make different 2D cross-sections depending on how it is cut.

3. Cone

A cone has one circular base and comes to a point.

• If you slice parallel to the base, you get a circle.
• If you slice straight down through the point and the base, you get a triangle.
• If you slice at a slant, you may get an oval.

4. Sphere

A sphere is a perfectly round solid, like a basketball.

No matter how you slice a sphere, the cross-section is always a circle.

The size of the circle can change. A slice through the middle makes the largest circle. A slice near the top or bottom makes a smaller circle.

5. Pyramid

A pyramid has a base and triangular faces that meet at one point.

• If you slice parallel to the base, the cross-section has the same shape as the base.
• If you slice through the top point, you often get a triangle.

For example, a square pyramid sliced parallel to its base makes a smaller square.

6. Triangular Prism

A triangular prism has two triangular bases and rectangular side faces.

• If you slice parallel to the triangular bases, the cross-section is a triangle.
• If you slice parallel to a rectangular face, the cross-section is a rectangle.

Important pattern to remember

• A slice parallel to a face or base usually gives the same shape as that face or base.
• Changing the direction of the slice can change the cross-section.
• The cross-section is always flat, so it is always a 2D shape.

How to identify a cross-section

1. Name the 3D solid.
2. Notice which way it is sliced.
3. Ask: Is the slice parallel to a face or base?
4. Picture the flat cut surface.
5. Name the 2D shape you would see.

Worked Example 1

A cylinder is sliced parallel to its circular base. What is the cross-section?

Step 1: The solid is a cylinder.

Step 2: The slice is parallel to the base.

Step 3: The base of a cylinder is a circle.

Answer: The cross-section is a circle.

Worked Example 2

A rectangular prism is sliced parallel to one of its side faces. What shape is the cross-section?

Step 1: The solid is a rectangular prism.

Step 2: The slice is parallel to a side face.

Step 3: A side face of a rectangular prism is a rectangle.

Answer: The cross-section is a rectangle.

Worked Example 3

A cone is sliced straight down through its top point and through the base. What is the cross-section?

Step 1: The solid is a cone.

Step 2: The slice goes through the point and the base.

Step 3: That kind of slice creates a flat shape with three sides.

Answer: The cross-section is a triangle.

Worked Example 4

A sphere is sliced near its top. What shape is the cross-section?

Step 1: The solid is a sphere.

Step 2: Any slice of a sphere makes a circle.

Step 3: Since the slice is near the top, the circle will be smaller than the middle slice.

Answer: The cross-section is a circle.

Comparing solids and cross-sections

• Cube  square cross-section when sliced parallel to a face
• Rectangular prism  rectangle cross-section when sliced parallel to a face
• Cylinder  circle or rectangle
• Cone  circle or triangle
• Sphere  circle only
• Pyramid  often the same shape as the base, or a triangle
• Triangular prism  triangle or rectangle

Common mistakes to avoid

• Do not name the 3D solid as the answer. A cross-section is a 2D shape, not a 3D figure.
• Do not forget that the direction of the slice matters.
• Do not assume one solid has only one cross-section. Many solids can make different shapes.

Quick check questions

• If a cube is sliced parallel to a face, what shape do you get? Square
• If a cylinder is sliced straight up and down, what shape can you get? Rectangle
• If a cone is sliced parallel to its base, what shape do you get? Circle
• If a sphere is sliced anywhere, what shape do you get? Circle

Summary

A cross-section is the flat 2D shape made when a 3D solid is sliced. The shape of the cross-section depends on the solid and the direction of the slice.

Remember: slices parallel to a base or face often make the same shape as that base or face. By practicing with cubes, prisms, cylinders, cones, pyramids, and spheres, you can get better at visualizing the 2D shapes hidden inside 3D solids.

Surface Area Foundations means finding the total area on the outside of a 3D shape.

Think about wrapping a gift box with paper. The amount of paper needed to cover the whole box is like the surface area of the box.

To find surface area, we break a 3D figure into its flat faces. These flat faces make a net. Then we find the area of each face and add them together.

In this lesson, we will learn how to:

• understand what surface area means,
• use nets to see all the faces of a 3D figure,
• find the area of each face,
• add the face areas to get total surface area.

Important idea: Surface area is measured in square units, such as square centimeters \\(cm^2\\) or square inches \\(in^2\\).

Step 1: Know the difference between 2D and 3D

A 2D shape is flat, like a rectangle or square. A 3D figure has length, width, and height, like a box or cube.

Surface area belongs to 3D figures, but we find it by using the areas of their 2D faces.

Step 2: Understand faces

A face is a flat side of a 3D figure. For example:

• A cube has 6 square faces.
• A rectangular prism has 6 rectangular faces.

When we unfold a 3D figure into a flat pattern, we get a net.

Step 3: Find the area of each face

Before finding surface area, we need to remember area formulas for flat shapes.

• Area of a rectangle: \\(A=l \times w\\)
• Area of a square: \\(A=s \times s\\)

Then add all the face areas:

$$\text{Surface Area} = \text{sum of the areas of all faces}$$

Step 4: Look for matching faces

Many 3D figures have faces that are the same size.

For example, a rectangular prism has:

• a top and bottom that match,
• a front and back that match,
• a left side and right side that match.

This helps us work faster because we can find one face area and count it twice.

Cube surface area

A cube has 6 equal square faces. If each side is \\(s\\), then one face has area:

$$s \times s = s^2$$

Since there are 6 faces, the total surface area is:

$$6 \times s^2$$

Rectangular prism surface area

A rectangular prism has 3 pairs of matching faces.

If the length is \\(l\\), width is \\(w\\), and height is \\(h\\), then the face areas are:

• top and bottom: \\(l \times w\\)
• front and back: \\(l \times h\\)
• left and right: \\(w \times h\\)

Add both of each pair:

$$2(l \times w) + 2(l \times h) + 2(w \times h)$$

You do not need to memorize this right away. You can always use the net and add the faces one by one.

Worked Example 1: Surface area of a cube

A cube has side length \\(4\\) cm. Find the surface area.

Step 1: Find the area of one face.

$$4 \times 4 = 16$$

So one face has area \\(16\,cm^2\\).

Step 2: Multiply by 6 because a cube has 6 faces.

$$6 \times 16 = 96$$

Answer: The surface area is \\(96\,cm^2\\).

Worked Example 2: Rectangular prism with different side lengths

A box is \\(5\\) cm long, \\(3\\) cm wide, and \\(2\\) cm high. Find the surface area.

Step 1: Find the three different face areas.

• Top/bottom: \\(5 \times 3 = 15\\)
• Front/back: \\(5 \times 2 = 10\\)
• Left/right: \\(3 \times 2 = 6\\)

Step 2: Count each pair twice.

$$2(15) + 2(10) + 2(6)$$

$$30 + 20 + 12 = 62$$

Answer: The surface area is \\(62\,cm^2\\).

Worked Example 3: Using a net

A rectangular prism has these face areas in its net:

• 2 faces of \\(12\,in^2\\)
• 2 faces of \\(8\,in^2\\)
• 2 faces of \\(6\,in^2\\)

Find the surface area.

Add all 6 faces:

$$12 + 12 + 8 + 8 + 6 + 6 = 52$$

Answer: The surface area is \\(52\,in^2\\).

This example shows that sometimes the face areas are already given. Then you just add them.

Worked Example 4: Finding a missing total from the dimensions

A storage box is \\(7\\) in long, \\(4\\) in wide, and \\(3\\) in high. Find the surface area.

Step 1: Find each type of face.

• Top/bottom: \\(7 \times 4 = 28\\)
• Front/back: \\(7 \times 3 = 21\\)
• Left/right: \\(4 \times 3 = 12\\)

Step 2: Double each one and add.

$$2(28) + 2(21) + 2(12)$$

$$56 + 42 + 24 = 122$$

Answer: The surface area is \\(122\,in^2\\).

Tips for success

• Draw the net if the 3D figure feels confusing.
• Label each face with its dimensions.
• Find area carefully using multiplication.
• Add all faces to get the total outside area.
• Check units: surface area must be in square units.

Common mistakes to avoid

• Do not add the side lengths. That would be perimeter, not surface area.
• Do not forget hidden faces like the bottom or the back.
• Do not use plain units like cm or in. Use \\(cm^2\\) or \\(in^2\\).
• Do not count a face only once when there are matching pairs.

How to think about surface area

Imagine painting all the outside faces of a box. Surface area tells how much space the paint covers.

Imagine cutting a box open and laying it flat. The area of that flat pattern, or net, is the same as the surface area of the box.

Quick practice questions

1. A cube has side length \\(2\\) ft. What is its surface area?
2. A rectangular prism has dimensions \\(6\\) cm by \\(2\\) cm by \\(1\\) cm. What is its surface area?
3. A net has face areas of \\(9, 9, 12, 12, 6, 6\\) square units. What is the total surface area?

Answers to quick practice

1. One face: \\(2 \times 2 = 4\\), so total is \\(6 \times 4 = 24\,ft^2\\).
2. $$2(6 \times 2) + 2(6 \times 1) + 2(2 \times 1) = 24 + 12 + 4 = 40\,cm^2$$
3. $$9+9+12+12+6+6=54$$ square units

Summary

Surface area is the total area on the outside of a 3D figure. We find it by looking at all the flat faces, finding each face area, and adding them together.

Nets are helpful because they show every face clearly. If you can find the area of rectangles and squares, you can build a strong foundation for finding surface area.