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

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

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?

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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

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