# How can I find the volume of a rectangle from its surface area? [closed]

The area of the base of an open rectangular box is 120cm2. The front has an area of 96cm2 and the side has an area of 100cm2. What is the volume of the box?

So I was given the surface area of three of the sides of the rectangular prism and I'm trying to figure out how to get the width, height and length only using the surface areas provided to me. I've done some trial and error but no luck. Is there any way I can go about this differently? Thanks!

Area of rectangle is length x height. Volume of rectangular prism is length x width x height.

## closed as off-topic by Alex Provost, mrtaurho, José Carlos Santos, Leucippus, ShaileshMar 18 at 0:56

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• Welcome to Math Stack Exchange. How do you calculate the area of a rectangle and the volume of a rectangular box? – J. W. Tanner Mar 17 at 5:16
• Area of rectangle is length x height. Volume of rectangular prism is length x width x height. – scratch Mar 17 at 5:18
• Welcome to MSE. Please add your thoughts to the question and put some effort in! – Rhys Hughes Mar 17 at 5:18
• @scratch you should edit your question and add that information – Rhys Hughes Mar 17 at 5:19
• @scratch: Put it this way. Let's say the lengths of sides of the box are $a$, $b$, and $c$. What do you then know about $a$, $b$, and $c$ from the areas given, and what is the volume of the box? – J. W. Tanner Mar 17 at 5:23

## 2 Answers

Say the sides of the box are $$a, b,$$ and $$c$$.

Then we have $$ab=120, ac=96$$, and $$bc=100$$ cm$$^2$$.

The volume is $$abc$$ cm$$^3$$.

Note that $$120\times96\times100$$ = $$ab\times ac\times bc = (abc)^2,$$ so

$$abc=\sqrt{120\times96\times100}=\sqrt{4\times30\times16\times6\times100}=$$

$$2\times 4\times 10\times\sqrt{5\times6\times6}=80\times6\sqrt5=480\sqrt5$$ cm$$^3$$.

• Thanks! So the answer must be in this case 1073.312629 cm3. – scratch Mar 17 at 5:38
• Yes (approximately) – J. W. Tanner Mar 17 at 5:39

Let the length, height, width be $$L, H, W$$ respectively.

Then you have:

$$LW=120\to L=\frac{120}{W}$$ $$LH=96\to L=\frac{96}{H}$$ $$WH=100$$

Notice from the first two that: $$\frac{120}{W}=\frac{96}{H}\to\frac WH = \frac{120}{96}=\frac 54$$

Use this with the third statement ($$WH=100$$) to find $$W$$ and $$H$$. Then use either of the first two to find $$L$$, armed with this information.

• For reference, I achieve that $$H=\sqrt{80}=4\sqrt5; W=5\sqrt 5; L=\frac{24}{5}\sqrt5$$ – Rhys Hughes Mar 17 at 5:35
• so we agree $HWL=480\sqrt 5$ – J. W. Tanner Mar 17 at 15:40
• Indeed we do, I actually didn't realise that your method (I.e. one so direct) was a possibility, so it was nice to see. – Rhys Hughes Mar 17 at 17:35