Volume of a rectangular prism's walls

Question

Sorry if this is an obvious question... I have been trying to figure this out for a little while and come up with nothing...

If I have a rectangular prism, say $5\times10\times12$ meters, it has a surface area of $460\ m^2$ ...

If the prism has walls that are $0.25$ meters thick (extending inwards)... how do I find the volume of those walls?

Thanks,
Mike

Answer 1

$$V=5\times10\times12-(5-.5)(10-.5)(12-.5)$$

This amounts to taking the difference in volume between the original rectangular prism and the inside rectangular prism.

Note that if we extend the wall a distance $x$ inward, then each side length of the interior prism's walls will differ from the original prism's walls by $2x$.

Answer 2

Good question!

So, let's "grow" the walls inside the prism. The volume of your walls would be $460 \times 0.25 = 115$ cubic meters, right?

Nope! You double-counted each of the edge volumes. Two walls grew into each edge.

So we subtract out one of those two edge volumes, which is

$$4 \times (5 + 10 + 12) \times 0.25 \times 0.25 = 6.75$$ cubic meters, leaving $115 - 6.75 = 108.25$ cubic meters, and we're done, right?

Wrong again! By taking out all of the edge volumes, we completely removed the corners. Each corner is the intersection of three edges, after all.

So, we finally need to add the corner volumes back in: $8 \times 0.25 \times 0.25 \times 0.25 = 0.125$ cubic meters, leaving the total volume of the walls $108.375$ cubic meters.

Now we're done.

This matches the answer that you get when you use Peter's method above. Calculate the volume of the outer box, and subtract from it the volume of the inner box:

$$V_{walls} = V_{outer} - V_{inner} = 5 \times 10 \times 12 - (5 - 0.5)(10-0.5)(12-0.5) = 108.375.$$

Answer 3

The volume of the walls is equivalent to the volume enclosed by the walls subtracted from the total volume of the prism. The volume of the prism is $600$ cubic meters, and the volume of the closed area is equal to $ (5-0.5)(10-0.5)(12-0.5) = 491.625 $ cubic meters because 0.25 meters is cut off from each side of any given edge. Subtracting we yield $ 600 - 491.625 = 108.375 $ cubic meters as the volume of the walls.