Surface Area of a Prism Examples in Math

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.

Answer

SA =94= 94 cm².

First step

1
Step 1: For a rectangular prism, the surface area formula is SA=2(lw+lh+wh)SA = 2(lw + lh + wh), which fits the general form SA=2B+PhSA = 2B + Ph for a rectangular base.

Full solution

  1. 2
    Step 2: Calculate each face pair: lw=5×3=15lw = 5 \times 3 = 15; lh=5×4=20lh = 5 \times 4 = 20; wh=3×4=12wh = 3 \times 4 = 12.
  2. 3
    Step 3: SA=2(15+20+12)=2×47=94SA = 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.

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 ss: A=332s2A = \frac{3\sqrt{3}}{2}s^2.)

Example 3

easy
Find the surface area of a cube with side length 55 cm.

Example 4

easy
A rectangular prism has dimensions 2×4×52 \times 4 \times 5 cm. Find its surface area.

Example 5

easy
Find the lateral surface area of a rectangular prism with base perimeter 2020 cm and height 77 cm.

Example 6

easy
A cube has surface area 216216 cm2^2. Find the edge length.

Example 7

easy
A rectangular prism has dimensions 3×3×83 \times 3 \times 8. Find its surface area.

Example 8

easy
A cube of edge 1010 cm: find the area of one face and the total surface area.

Example 9

medium
A triangular prism has equilateral triangular bases of side 44 cm (base area 434\sqrt{3} cm2^2) and length 99 cm. Find the total surface area.

Example 10

medium
A rectangular prism has volume 6060 cm3^3 with a square base of side 33 cm. Find its surface area.

Example 11

medium
A triangular prism has right-triangle bases with legs 66 and 88 (hypotenuse 1010) and length 1212. Find its total surface area.

Example 12

medium
A box has a square base of side 44 cm and total surface area 112112 cm2^2. Find its height.

Example 13

medium
An open-top rectangular box (no lid) is 8×5×48 \times 5 \times 4. Find the surface area of the material used.

Example 14

medium
A trapezoidal prism has a trapezoidal base with parallel sides 55 and 77 and height 44 (between them). The other two sides of the trapezoid are each 55. The prism has length 1010. Find the lateral surface area.

Example 15

medium
A cereal box is 20×6×3020 \times 6 \times 30 cm. Find the total surface area of cardboard needed.

Example 16

medium
Two cubes of edge 22 are joined face-to-face to make a 2×2×42 \times 2 \times 4 prism. Find its surface area.

Example 17

hard
A regular hexagonal prism has side length 33 cm and height 1010 cm. The hexagonal base has area 332⋅9=2732\frac{3\sqrt{3}}{2} \cdot 9 = \frac{27\sqrt{3}}{2} cm2^2. Find the total surface area.

Example 18

hard
A box has volume 7272 cm3^3 and a square base of side 33 cm. Find its surface area.

Example 19

hard
A pentagonal prism has a regular pentagonal base of perimeter 2020 cm and area 27.527.5 cm2^2. The prism height is 88 cm. Find the total surface area.

Example 20

hard
A 2×2×42 \times 2 \times 4 rectangular prism is cut out from one full corner of a 4×4×44 \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 55 and 1212 (hypotenuse 1313) and prism length 2020. Find the total surface area.

Example 22

challenge
Among all closed rectangular boxes with surface area 9696 cm2^2, which dimensions maximize the volume?

Example 23

challenge
A 5×5×55 \times 5 \times 5 cube is built from 125125 unit cubes. A 1×1×51 \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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Background Knowledge

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

areasurface area