Find the volume of the given oblique rectangular prism.
An oblique prism is one in which the bases are not vertically aligned. We know how to calculate the volume of a right prism. It’s equal to the base area 𝐵 multiplied by the perpendicular height ℎ. But can we apply this formula to calculate the volume of an oblique prism? Well, in fact, we can.
Picture two piles of identical coins. In one pile, the coins are stacked directly on top of each other. In the other, the stack has been pushed slightly so that now it’s leaning to the side. Both of these piles have the same cross-sectional area. And they have the same perpendicular height. They also have the same volume as they’re identical coins.
This is an illustration of a principle called Cavalieri’s principle, which tells us that if two solids have the same height ℎ and the same cross-sectional area 𝐵 at every level, then they have the same volume. What all of this means is that, in order to calculate the volume of this oblique rectangular prism, we can in fact treat it as if it were a right prism.
We calculate first the area of the rectangular base and then multiply it by the perpendicular height of 3.2 meters. So we have that the volume is equal to 2.7 multiplied by four, for the area of the rectangular base, multiplied by 3.2.
Now the brackets in this calculation are actually mathematically unnecessary. I’ve just included them so we can see the parts of the calculation that make up the base and the part that makes up the height. Evaluating this gives that the volume of this oblique rectangular prism is 34.56 cubic meters.