Volume of a Cuboid

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.

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

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

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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?

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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?

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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

$$\begin{aligned} a'b'&=4ab\\ b'c'&=4bc\\ c'a'&=4ca. \end{aligned}$$

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?

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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

$$1 \times 2 \times 3 =6.\ _\square$$