Video: Volumes of Oblique Prisms

Lauren McNaughten

Learn how to calculate the volume of an oblique prism based on Cavalieri’s principle. Apply your knowledge to calculating the volumes of oblique rectangular, square, and hexagonal prisms and oblique cylinders.


Video Transcript

In this video, we’re going to look at how to calculate the volume of oblique prisms. So first we need to be clear what is meant by the term oblique prism. And there’s a diagram on the screen here to help us understand this. You’re perhaps already be familiar with what’s called “a right prism,” which I’ve drawn on the left of the screen. And what you notice about this is it for the right prism, the lateral faces are perpendicular to the bases. So I’ve marked one of the lateral faces in orange and then the base in green. And you can see that they are perpendicular to each other or at a right angle, which is where the name “right prism” comes from.

Now if you look at the oblique prism and if I do the same shading again, so there’s the base marked in green and one of the lateral faces shaded in orange, you’ll see that this time they are not perpendicular to each other. So there’s the difference between these two types of prism. In the right prism as I said the lateral faces are perpendicular to the bases, whereas in the oblique prism that is not the case. Well it also means that in the right prism the lateral faces are rectangles, whereas in the oblique prism the lateral faces are parallelograms. Now you have already seen how to calculate the volume of a right prism. This video we’re focusing on the oblique prisms. So in order to think about the volume of oblique prisms, we need to rely on a principle called “Cavalieri’s principle.” And it says the following: if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

So take a look at the diagram. These two prisms, one is a right prism and one is an oblique prism. They have the same base area marked in green. And because they’re prisms, they will have that same area at every level throughout their height. And they also have the same height marked in ℎ. Now an important thing to know is that it is the perpendicular height. So in the case of the right prism, that’s just its usual height and in the case of the oblique prism is that height that’s perpendicular to the base; you see I’ve drawn a right angle in a continuation of that baseline there.

So Cavalieri’s principle tells us that the volume of these two prisms are equal. One way perhaps to help visualize this is to think about a stack of coins. So in one case, we have these coins stacked directly on top of each other like in the right prism, whereas in the other, we have them in a sort of diagonal stack if you can get them to balance like that — like in an oblique prism. And if you were to do this yourself, you’d see that the height of those two stacks of coins is the same. And of course the volume is the same because they’re just the same coins arranged in a different formation. So that gives a helpful physical demonstration of this principle.

What all of this means then, in the case of calculating the volume of an oblique prism is we can essentially treat them exactly the same as we do right prisms by working out the cross-sectional area and then multiplying it by the height of the prism. So we can treat these two types of prisms in exactly the same way. The only caveat with the oblique prism is we need to make sure we’re using the perpendicular height and not a sloping height.

So let’s look at applying this to our first question. We’re asked to calculate the volume of the oblique rectangular prism below. So remember from the previous discussion that the volume of this oblique prism will be the area of the cross section or the base multiplied by the height. So I’m going to use 𝐵 to represent base and ℎ to represent height in my formulae here. So looking at the prism then, the base of it or in this case the top that I’ve shaded in is a rectangle with dimensions of two and five. So that would be fine to work out this area. We then just need to think about the height of the prism carefully. Because you’ve actually been given two different heights: we’ve been given the six meters, which is that sloping height, and we’ve been given four meters, which is the perpendicular height. And remember is the perpendicular height that we need. So in this question, we’ve actually been given more information than is necessary in order to test that we truly understand the method for calculation a volume of an oblique prism. So our calculation then, the volume is the base area. Well as we said that’s a rectangle with dimensions of two and five. So two times five multiplied by the height and we must use that measurement of four meters, the perpendicular height. So working this out gives us a volume of forty cubic meters for this oblique rectangular prism.

Okay, the next question says calculate the volume of an oblique hexagonal prism with a perpendicular height of ten centimetres and a base area of sixty-five centimetres squared. So let’s recall the volume formula that we need. And of course it’s this formula that the volume is equal to the base multiplied by the perpendicular height. So we’ve been given both of those measurements; we just need to substitute them into this formula. So in the case of this hexagonal prism, the base area is sixty-five; the perpendicular height is ten. So to calculate the volume, we’re multiplying sixty-five by ten. And this gives us an answer then of six hundred and fifty cubic centimetres.

The next question asks us to find the volume of an oblique square prism with height seven point two millimetres and base edges of length four point five millimetres. So as always we need to recall that volume formula, which is the base area multiplied by the height. Now we’ve got the height; it’s seven point two millimetres. And because this is an oblique square prism, we can calculate the area of the base by multiplying these two sides together, so four point five times four point five. So our calculation of the volume then is just four point five multiplied by four point five to give the base area then multiplied by seven point two, which is the height. This gives us an answer then of one hundred and forty-five point eight millimetres cubed. So in each of these questions, the only real consideration so far has been the shape of the base because of course that affects the calculation that we do in order to find its area. In the case of a square or rectangle, it’s relatively straightforward. Remember of course if it’s a triangle, you have to divide by two or if it’s another type of two-dimensional shape, you just have to record the relevant formula for calculating its area.

Right, the final question asks us to find the volume of the oblique cylinder shown. So we need of course our volume formula. And in the case of the cylinder, the base is of course a circle. So we also need to recall the formula for finding the area of circle, which remember is 𝜋𝑟 squared, where 𝑟 represents the radius of the circle. So using those two formula, let’s calculate the volume of this oblique cylinder. So base area first of all is 𝜋𝑟 squared. Well if we look at the diagram, we haven’t actually been given the radius; we’ve been given the diameter of the circle, which is six centimetres. So we need to halve it in order to find the radius. So we have 𝜋 multiplied by three squared for the area of the base. Then we need to multiply by the height, so multiplied by five. This gives us an answer then of forty-five 𝜋. And we could leave our answer like that if we didn’t have a calculator or we wanted an exact answer or indeed if it was requested, but I’ll go on and evaluate this answer as a decimal. And this gives me an answer then of one hundred and forty-one point four cubic centimetres and that’s been rounded to one decimal place.

So to summarize then, when working with oblique prisms due to Cavalieri’s principle, you can treat them in the same way that you do right prisms, and you can calculate their volumes by working out the area of the base and then multiplying by the height. Just make sure you are using the perpendicular height as opposed to any kind of slant height of the prism.

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