Surface Area of a Prism Examples: 25 Problems with Answers

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

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 faces of a prism, found by adding the areas of the two bases and all lateral (side) faces.

Imagine unfolding a cereal box and laying it flat—you get a net of six rectangles. The surface area is the total area of that flattened cardboard. For any prism, you always have two identical bases plus a 'belt' of rectangles wrapped around the middle.

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 of a prism is the total area of its two bases plus all the side faces.

Common stuck point: The procedure for surface area of a prism is the easy part; the trap is counting only one base. Asking "Am I adding the areas of every outer face of a prism (not filling its inside)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I adding the areas of every outer face of a prism (not filling its inside)?

Worked Examples

Example 1

easy

A rectangular prism (box) has length 5 cm, width 3 cm, and height 4 cm. Find its surface area.

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

Answer

= 94

cm².

First step

Step 1: For a rectangular prism, the surface area formula is

SA = 2(lw + lh + wh)

, which fits the general form

SA = 2B + Ph

for a rectangular base.

Full solution

2
Step 2: Calculate each face pair: $lw = 5 \times 3 = 15$ ; $lh = 5 \times 4 = 20$ ; $wh = 3 \times 4 = 12$ .
3
Step 3: $SA = 2(15 + 20 + 12) = 2 \times 47 = 94$ cm².

The surface area of a rectangular prism consists of 3 pairs of congruent rectangles. Each pair has area equal to the product of two dimensions. The total is twice the sum of the three face areas. This is the most common prism in everyday life (cereal boxes, shipping boxes, etc.).

Example 2

medium

A triangular prism has a triangular base with base 6 cm and height 4 cm, and the prism's length is 10 cm. The three sides of the triangular base are 5, 5, and 6 cm. Find the total surface area.

Practice Problems

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

Example 1

easy

A cube has side length 7 cm. Find its surface area.

Cube with side length 7 cm

Example 2

hard

A prism has a regular hexagonal base with side length 4 cm. The prism has height 10 cm. Find the surface area. (Area of regular hexagon with side

s

A = \frac{3\sqrt{3}}{2}s^2

Example 3

easy

Find the surface area of a cube with side length

5

cm.

Cube with side length 5 cm

Example 4

easy

A rectangular prism has dimensions

2 \times 4 \times 5

cm. Find its surface area.

Rectangular prism: 2 × 4 × 5 cm

Example 5

easy

Find the lateral surface area of a rectangular prism with base perimeter

20

cm and height

7

cm.

Example 6

easy

A cube has surface area

216

^2

. Find the edge length.

Cube with unknown edge s; SA

Example 7

easy

A rectangular prism has dimensions

3 \times 3 \times 8

. Find its surface area.

Rectangular prism: 3 × 3 × 8

Example 8

easy

A cube of edge

10

cm: find the area of one face and the total surface area.

Cube with edge 10 cm

Example 9

medium

A triangular prism has equilateral triangular bases of side

4

cm (base area

4\sqrt{3}

^2

) and length

9

cm. Find the total surface area.

Example 10

medium

A rectangular prism has volume

60

^3

with a square base of side

3

cm. Find its surface area.

Square-base prism: side 3 cm, height h unknown; V

Example 11

medium

A triangular prism has right-triangle bases with legs

6

and

8

(hypotenuse

10

) and length

12

. Find its total surface area.

Example 12

medium

A box has a square base of side

4

cm and total surface area

112

^2

. Find its height.

Square-base box: side 4 cm, height h = ?

Example 13

medium

An open-top rectangular box (no lid) is

8 \times 5 \times 4

. Find the surface area of the material used.

Open-top box: 8 × 5 × 4 (no lid)

Example 14

medium

A trapezoidal prism has a trapezoidal base with parallel sides

5

and

7

and height

4

(between them). The other two sides of the trapezoid are each

5

. The prism has length

10

. Find the lateral surface area.

Example 15

medium

A cereal box is

20 \times 6 \times 30

cm. Find the total surface area of cardboard needed.

Cereal box: 20 × 6 × 30 cm

Example 16

medium

Two cubes of edge

2

are joined face-to-face to make a

2 \times 2 \times 4

prism. Find its surface area.

Two 2×2×2 cubes joined → 2×2×4 prism

Example 17

hard

A regular hexagonal prism has side length

3

cm and height

10

cm. The hexagonal base has area

\frac{3\sqrt{3}}{2} \cdot 9 = \frac{27\sqrt{3}}{2}

^2

. Find the total surface area.

Example 18

hard

A box has volume

72

^3

and a square base of side

3

cm. Find its surface area.

Square-base box: side 3 cm, h unknown; V

Example 19

hard

A pentagonal prism has a regular pentagonal base of perimeter

20

cm and area

27.5

^2

. The prism height is

8

cm. Find the total surface area.

Example 20

hard

2 \times 2 \times 4

rectangular prism is cut out from one full corner of a

4 \times 4 \times 4

cube (the removed prism shares three faces with the cube). Find the total surface area of the remaining solid.

Example 21

hard

A right triangular prism has base legs

5

and

12

(hypotenuse

13

) and prism length

20

. Find the total surface area.

Example 22

challenge

Among all closed rectangular boxes with surface area

96

^2

, which dimensions maximize the volume?

Example 23

challenge

5 \times 5 \times 5

cube is built from

125

unit cubes. A

1 \times 1 \times 5

column of unit cubes is removed straight through the cube. Find the surface area of the resulting solid.

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Related Concepts

Area Surface Area Surface Area of a Cylinder Nets

Background Knowledge

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

areasurface area