Surface Area Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Surface Area.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The total area of all the faces or surfaces that enclose a three-dimensional object, measured in square units.

How much wrapping paper would you need to completely cover every face of a gift box?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Surface area is the total area of all the outside faces that enclose a 3D solid, added together in square units.

Common stuck point: The procedure for surface area is the easy part; the trap is multiplying all three dimensions. Asking "Am I adding up the areas of all the outside faces of a solid?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I adding up the areas of all the outside faces of a solid?

Worked Examples

Example 1

easy

Find the surface area of a rectangular prism with length

4

cm, width

3

cm, and height

5

cm.

Rectangular prism with l = 4 cm, w = 3 cm, h = 5 cm

Answer

SA = 94 \text{ cm}^2

First step

A rectangular prism has 6 faces forming 3 pairs of identical rectangles. Label the pairs:

l \times w

(top/bottom),

l \times h

(front/back),

w \times h

(left/right). Total:

SA = 2(lw + lh + wh)

Full solution

2
Substitute $l = 4$ cm, $w = 3$ cm, $h = 5$ cm: calculate each pair — $lw = 12$ , $lh = 20$ , $wh = 15$ .
3
Compute: $SA = 2(12 + 20 + 15) = 2(47) = 94$ cm². Each pair of faces contributes twice to the total surface.

Surface area is the total area of all faces of a 3D shape. For a rectangular prism, compute the area of each distinct face and double it (since opposite faces are congruent).

Example 2

medium

Find the surface area of a cylinder with radius

4

cm and height

7

cm. Leave your answer in terms of

\pi

Cylinder with r = 4 cm and h = 7 cm

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the surface area of a cube with side length

6

cm.

Cube with side length 6 cm

Example 2

medium

A cube has surface area

150

cm². Find the length of one edge.

Example 3

easy

Find the surface area of a cube with side length

3

Cube with side length 3

Example 4

easy

What is surface area measured in: square units or cubic units?

Example 5

easy

How many faces does a rectangular box (cuboid) have?

Example 6

easy

A gift box needs to be wrapped. Are you finding its surface area or its volume?

Example 7

easy

Find the surface area of a cube with side

1

Example 8

easy

A rectangular box has faces of areas

6

6

10

10

15

, and

15

. Find its total surface area.

Example 9

easy

Which uses square units: the amount of paint to cover a ball, or the air inside it?

Example 10

easy

A cube has surface area

24

. Find the area of one face.

Example 11

medium

Find the surface area of a rectangular box with length

4

, width

3

, height

2

Rectangular box with l = 4, w = 3, h = 2

Example 12

medium

Find the surface area of a cylinder with radius

3

and height

5

(in terms of

\pi

Cylinder with r = 3 and h = 5

Example 13

medium

Two cubes have surface areas

24

and

96

. Find the ratio of their side lengths.

Example 14

medium

A cube and a sphere have the same volume. Which has the larger surface area? (State the principle.)

Example 15

medium

A cube of side

4

is painted, then cut into unit cubes. Two opposite faces' worth of painted area — what is the total painted surface area of the original cube?

Example 16

medium

Find the surface area of a sphere with radius

3

(in terms of

\pi

Sphere with r = 3

Example 17

medium

A net is a flat unfolding of a 3D shape. How does a net help find surface area?

Example 18

medium

Find the surface area of a square-based pyramid with base side

6

and slant height

5

Example 19

challenge

A cube of side

3

is cut into 27 unit cubes. What is the total surface area of all 27 small cubes combined, and how does it compare to the original?

Example 20

challenge

A closed cylinder has radius

r

and height equal to its diameter (

h = 2r

). Express its surface area in terms of

r

and

\pi

, simplified.

Example 21

challenge

Why does a small animal lose body heat faster than a large one, in terms of surface-area-to-volume ratio?

Example 22

challenge

A cube and a sphere both have surface area

S

. Show which encloses more volume, and name the principle.

Example 23

easy

Find the surface area of a cube with side

2

cm.

Cube with side length 2 cm

Example 24

easy

Find the surface area of a rectangular box with length

5

, width

4

, height

2

Rectangular box with l = 5, w = 4, h = 2

Example 25

easy

A cube has surface area

54

^2

. Find the edge length.

Example 26

easy

Find the surface area of a sphere with radius

2

, in terms of

\pi

Sphere with r = 2

Example 27

easy

A cone has slant height

5

and base radius

3

. Find its lateral surface area in terms of

\pi

Cone with r = 3 and slant height = 5

Example 28

easy

Find the total surface area of a square pyramid with base side

4

and slant height

6

Example 29

medium

A closed cylinder has radius

2

cm and height

10

cm. Find its surface area in terms of

\pi

Closed cylinder with r = 2 cm and h = 10 cm

Example 30

medium

A sphere has surface area

100\pi

^2

. Find its radius.

Sphere with surface area 100π cm² — find r

Example 31

medium

A cone has radius

6

and height

8

. Find its total surface area in terms of

\pi

Cone with r = 6, h = 8, slant height = 10

Example 32

medium

A rectangular tank (open top) measures

4 \times 3 \times 2

m. Find the area of the inside walls plus bottom that need waterproofing.

Example 33

medium

Find the surface area of a cube whose space diagonal is

\sqrt{75}

Example 34

medium

A cylindrical can has radius

3

cm and height

12

cm. Find the lateral surface area in terms of

\pi

Cylinder with r = 3 cm and h = 12 cm

Example 35

medium

A square pyramid has base side

10

cm and height

12

cm. Find the total surface area.

Example 36

medium

A cube of side

4

is cut into

64

unit cubes. What is the total surface area of all unit cubes combined?

Example 37

hard

A cylinder has the same surface area as a cube with edge

6

. If the cylinder has radius

3

, find its height in terms of

\pi

Example 38

hard

Find the surface area of a regular tetrahedron with edge length

6

Example 39

hard

A cylinder is inscribed in a cube of edge

6

(so it just fits, tangent to all four side faces). Find the lateral surface area of the cylinder in terms of

\pi

Example 40

hard

A solid is made by stacking a cube of side

4

on top of a cube of side

6

(the small cube is centered on the top face of the big one). Find the total surface area.

Example 41

hard

A cone and a cylinder have the same radius

4

and same height

3

. Find the ratio of their lateral surface areas.

Example 42

challenge

A sphere is inscribed in a cube of side

6

. Find the ratio of the cube's surface area to the sphere's surface area, in simplest form.

Example 43

challenge

Among all closed cylinders with surface area

54\pi

, the maximum volume occurs when the height equals the diameter. Find that radius.

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Related Concepts

Area Volume Nets Sphere Surface Area

Background Knowledge

These ideas may be useful before you work through the harder examples.

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