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Find the volume of the given oblique rectangular prism.
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An oblique prism is one in which the bases are not vertically aligned.
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We know how to calculate the volume of a right prism.
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It’s equal to the base area 𝐵 multiplied by the perpendicular height ℎ.
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But can we apply this formula to calculate the volume of an oblique prism?
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Well, in fact, we can.
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Picture two piles of identical coins.
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In one pile, the coins are stacked directly on top of each other.
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In the other, the stack has been pushed slightly so that now it’s leaning to the side.
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Both of these piles have the same cross-sectional area.
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And they have the same perpendicular height.
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They also have the same volume as they’re identical coins.
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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.
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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.
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We calculate first the area of the rectangular base and then multiply it by the perpendicular height of 3.2 meters.
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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.
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Now the brackets in this calculation are actually mathematically unnecessary.
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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.
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Evaluating this gives that the volume of this oblique rectangular prism is 34.56 cubic meters.