Volume of a Rectangular Prism when given only the Surface Area

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So, in my equation, they ask you to find the volume of a rectangular prism when you only are given the surface area. As an example, one of the equations gave you a surface area of 240 yards squared, and no side measurements of any sort. How can I solve this if supposedly you need the length, width and height to find the volume?

Thank you for your time.

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3 Answers

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As the other answer suggests, if we don't place any restriction on what values the sides of a cuboid can take, there are way too many possibilities. We will assume the sides of the cuboid are all integers.

Given a cuboid of dimension $a \times b \times c$ where $a,b,c$ are integers, we have

$$\begin{align} \verb/Area/ &= 2(ab+bc+ca)\\ \verb/Vol/ &= abc\\ \end{align}$$ WOLOG, we will assume $a \le b \le c$ and rewrite the formula for area as $$(a+b)(a+c) = (ab+bc+ca)+a^2 = \frac12\verb/Area/+a^2 = 120+a^2$$ Since $$\verb/Area/ = 240 \implies 120 = (ab+bc+ca) \ge 3a^2 \implies 1 \le a \le 6$$

We have only $6$ cases to analysis. $$\begin{array}{clll} a = 1 &\implies 120+a^2 = 121 = 11\times 11 &\leadsto& (a,b,c,\verb/Vol/) = (1,10,10,100)\\ a = 2 &\implies 120+a^2 = 124 = 4 \times 31 &\leadsto& (a,b,c,\verb/Vol/) = (2,2, 29,116)\\ a = 3 &\implies 120+a^2 = 129 = 3 \times 43 &\leadsto& \verb/nothing/\\ a = 4 &\implies 120+a^2 = 136 = 2^3 \times 17 &\leadsto& (a,b,c,\verb/Vol/) = (4,4,13,208)\\ a = 5 &\implies 120+a^2 = 145 = 5 \times 29 &\leadsto& \verb/nothing/\\ a = 6 &\implies 120+a^2 = 156 = 2^2 \times 3 \times 13 &\leadsto& (a,b,c,\verb/Vol/) = (6,6,7,252) \end{array}$$

To summarize, if the sides of the cuboids are all integers, there are 4 possible volumes $100, 116, 208, 252$.

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You can't. For example, a surface area of 18 can have volume 4 (4x1x1), or volume 8 (2x2x2), or any other of infinitely many possibilities. You don't have enough information to solve the problem.

However, if you have additional information, like integer dimensions, then it is possible (although there may be multiple solutions). In this case, it would be hard to be any more efficient than to iterate through possibilities for the given surface area and checking the available volumes.

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I presume there is an assumption that the dimensions of the rectangular prism (box) are integers. Then if it is $a \times b \times c$, its surface area is $2(a b + a c + b c)$, i.e., the sum of the areas of the six faces.

Setting this expression to $240$ and solving leads to $a=1$, $b=10$, $c=10$, whence the volume is $100$.

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