# Volume 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. A cube is a cuboid whose faces are all squares.

An \(a \times b \times c \) cuboid has a volume of \( abc \) and a surface area of \( 2 ( ab+bc+ca) \).

## Examples

What is the volume of a \( 2 \times 3 \times 4 \) cuboid?

The volume is \( 2 \times 3 \times 4 = 24 \). \( _\square \)

The volume of an \( 8 \times 9 \times 10 \) cuboid is how many times the volume of a \( 3 \times 4 \times 5 \) cuboid?

The volume of an \( 8 \times 9 \times 10 \) cuboid divided by the volume of a \( 3 \times 4 \times 5 \) cuboid is

\[\frac{8 \times 9 \times 10 }{3 \times 4 \times 5}=2\times 3\times 2=12.\]

Thus, the answer is 12 times. \( _\square \)

The area of each of the 6 faces of a cuboid has grown by a factor of 4 times. Then the volume of the cuboid has grown by a factor of how many times?

Let \(a \times b \times c\) be the dimensions of the original cuboid. Then it has a volume of \( abc \) and a surface area of \( 2 ( ab+bc+ca) .\) Now, let \(a' \times b' \times c'\) be the dimensions of the enlarged cuboid. Then we have

\[\begin{align} a'b'&=4ab\\ b'c'&=4bc\\ c'a'&=4ca. \end{align}\]

Multiplying the three equations gives

\[(a'b'c')^2=64(abc)^2 \implies a'b'c'=8abc,\]

which implies that the new volume is 8 times the original. \( _\square \)

Suppose that an \(a\times b\times c\) cuboid has a surface area of 22, where \(a\ne b, b\ne c,\) and \(c \ne a.\) If \(a, b,\) and \(c\) are all integers, what is the volume of the cuboid?

We know that an \(a \times b \times c \) cuboid has a volume of \( abc \) and a surface area of \( 2 ( ab+bc+ca) .\) For this problem, it is given that \( 2 ( ab+bc+ca)=22,\) or

\[ ab+bc+ca=11. \qquad (1)\]

Without loss of generality, let \(a=1\) and \(b=2,\) then \((1)\) gives

\[1\times 2+2\times c+c\times 1=11 \Rightarrow c=3.\]

Since no other 3 distinct integers than 1, 2, and 3 can satisfy \((1),\) the volume of the cuboid is

\[1 \times 2 \times 3 =6.\ _\square\]

**Cite as:**Volume of a Cuboid.

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