# Surface Area of a Cuboid

## Summary

A (rectangular) cuboid is a closed box which comprises of 3 pairs of rectangular faces that are parallel to each other, and joined at right angles. It is also known as a right rectangular prism. It has 8 vertices, 6 faces, and 12 edges. The cube is a cuboid whose faces are all squares.

The figure above shows the developmental figure of a $a\times b\times c$ cuboid. Notice that the developmental figure consists of two $a\times b$ faces, two $b\times c$ faces, and two $c\times a$ faces. The surface area of a cuboid is the sum of the areas of these 6 faces. Hence the surface area of a $a\times b\times c$ cuboid is $2(ab+bc+ca)$.

## Examples

## What is the surface area of a $2 \times 3 \times 4$ cuboid?

The surface area is $2 ( 2 \times 3 + 3 \times 4 + 4 \times 2) = 52$. $_\square$

## There is a cuboid whose base is a square and height is 10 cm. If the area of one side face is $120\text{ cm}^2$, what is the total surface area of the cuboid?

Since the area of one side face is $120\text{ cm}^2$, the side length of the square base is $\frac{120}{10} = 12\text{ cm}$. Thus, the cuboid is a $12\text{ cm}\times 12\text{ cm}\times 10\text{ cm}$ cuboid. Hence, the surface area of the cuboid is

$2 \times (12 \times 12 + 12 \times 10 + 12 \times 10) = 768\ (\text{cm}^2).\ _\square$

## A cuboid's base has an area of $20\text{ cm}^2$, and a perimeter of 20 cm. If the cuboid's height is 6 cm, what is the surface area?

The sum of the areas of the side faces is (perimeter of base) $\times$ (height), and the surface area of the cuboid is (sum of areas of the side faces) + 2 $\times$ (area of base). Thus, the surface area of the cuboid is $20 \times 6 + 2\times 20 = 160\ (\text{cm}^2).\ _\square$

## Consider a cuboid whose base is a $3\text{ cm}\times4\text{ cm}$ rectangle. If the volume of the cuboid is $60\text{ cm}^3,$ what is the surface area?

An $a \times b \times c$ cuboid has a volume of $abc$ and a surface area of $2 ( ab+bc+ca)$. Since the cuboid's volume is $60\text{ cm}^3$ and the base is a $3\text{ cm}\times4\text{ cm}$ rectangle, the height is $\frac{60}{3 \times 4} = 5\text{ cm}.$ Then, the surface area of the cuboid is $2 \times ( 3 \times 4 + 3 \times 5 + 4 \times 5) = 94\ (\text{cm}^2).\ _\square$

**Cite as:**Surface Area of a Cuboid.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/surface-area-cuboid/