# Video: Volumes of Prisms

Derive a formula for calculating the volume of a prism: the area of cross-section times the depth of the prism. We will run through examples demonstrating the technique of using the formula to calculate the volume of prisms including cubes, rectangular prisms, and triangular prisms.

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### Video Transcript

In this video, we are going to look at how to calculate the volume of a prism.

Now, there are two words in that title that we need to be comfortable with. And the first of those is volume. The volume of a 3D shape, or a solid, remember, is the amount of space enclosed within that solid. Secondly, a prism, a prism is a particular type of three-dimensional shape. And it’s one in which the cross section of the solid is always the same. So, no matter where you cut the solid along its length, you always see exactly the same face, exactly the same shape, and exactly the same size.

Examples of prisms are things like cubes and cuboids. Or cylinders are types of prism. But you can also have prisms with any shape at all as the cross section. So, you could have a triangular prism, or a prism where the cross section is a parallelogram or a trapezium perhaps. So, let’s look at an example of calculating volume.

This question says, given that each of the small cubes in the solid below has a volume of one cubic centimetre, calculate the total volume of the solid.

So, if we know the volume of each of the individual cubes, then really it’s just a question of adding up how many cubes there are within this shape. But we don’t want to just go through and add them all up because that would take a very long time if it was a particularly large solid. So, we need to think about a slightly quicker way of doing it.

So, if we look at the base of this solid first of all, we can see that it has one, two, three, four, five, six cubes arranged in a row in this direction. And it has two cubes in each row in that direction. Which means that if I think about just the number of cubes, or the volume, of just this bottom layer of the solid, so the layer I have shaded in green, then it will be equal to six multiplied by two. It will be 12 centimetres cubed, or 12 cubic centimetres.

But I don’t want the volume of just one of these layers. I want the volume of this total solid. So, I need to think about the fact that there are, in fact, three layers of these. So, the total volume I need to multiply is 12 by three. So, that gives me a volume of 36 cubic centimetres for the total solid.

Now, in the previous example, we worked out the volume of that solid, which was a cuboid, just by counting how many little cubes were within it. But that’s not always a particularly practical way to answer such a question. So, let’s just think about a general formula that we can use to calculate the volume of any prism.

So, I’ve chosen to use a triangular prism for this. And we’ll just think about the method behind what we did last time. Essentially what we did was we worked out one layer of the prism, and then we multiplied it by how many there were. So, what that’s equivalent to, in terms of a general prism, is working out the area of the face on the front of the prism, so this constant cross section. And then, multiplying it by the height or the depth of the prism, which is essentially equivalent to how many layers of this shape you have.

So, this gives us a method, or a formula, for calculating the volume of any prism. We need to work out the area of its cross section first of all, whether that’s a square, a rectangle, a triangle, a circle, whatever it might be. And then, multiply it by the height of this prism. And so, this formula will work regardless of the type of prism that we’re given. So, let’s apply this to a couple of questions.

The first question says, calculate the volume of a cube of side length five centimetres.

So, in a cube, remember, all of the sides are the same length cause all of the faces are squares. So, this is what we’re interested in here, a cube with sides of length five. So, our formula tells us to work out the area of the cross section. Now, in the case of a cube, the cross section is a square. And the area of that is gonna be five multiplied by five. And then, we have to multiply by the height, or the depth, of the prism. And in the case of this cube, that’s also five centimetres. So, our calculation for the volume is five times five times five, or five cubed, which, of course, is 125 centimetres cubed. Just a note about units, they are cubic units because we’re talking about three-dimensional space, so centimetres cubed, metres cubed, millimetres cubed, and so on.

Right, the next question asks us to calculate the volume of the given cuboid. And we can see it’s got dimensions of five centimetres, 12 centimetres, and three centimetres.

So, to calculate the volume, we need the area of the cross section. And then, we need to multiply it by the depth, or the height, of the prism. Now, a cuboid is quite special in that whichever way you choose to slice it, whether it’s horizontal slices, vertical slices, or left-to-right, that cross section that you see is always constant in any direction. So, we can actually choose any of these faces to be our cross section, but I’m gonna choose the front of the prism. So, this face that I’ve marked in green.

So, then, my calculation for the volume, well, it’s the area of this cross section. So, it’s a rectangle with dimensions of five and three. So, that’s gonna be five multiplied by three. And then, I have to multiply by the depth, or the height, of the prism, so that’s the remaining measurement of 12 centimetres. So, my calculation for this cuboid is just five times three times 12. That gives me an answer then of 180 centimetres cubed for the volume of this cuboid.

So, our next question asks us to calculate the volume of the triangular prism shown.

So, for this prism, that constant cross section that you see wherever you slice it is a right-angled triangle. This triangle that’s on the front face of the prism here. So, to calculate the volume, remember, I need to do the area of the cross section. So, for the triangle, that’s base times height over two, or three times four divided by two. And then, I need to multiply it by the height, or the depth, of the prism, so that’s this measurement of eight centimetres.

So, I have three times four over two multiplied by eight. And if I then evaluate it, it gives me an answer of 48 centimetres cubed for the volume of this prism. So, it is just a question of thinking about what is the shape of the cross section of the prism you’re given. For a triangle, don’t forget to divide by two. That is a common error that people often make.

Right, our final question says, calculate the volume of the prism below.

Now, if you look at this prism, you will see that the front face, which is the constant cross section. So, this face that I’m shading in in green here. This face is a trapezium because it’s got one pair of parallel sides. And in fact, it’s a right-angled trapezium. So, we’re going to calculate the volume in exactly the same way, which means working out the area of this front face and then multiplying by the depth. But we need to recall how to calculate the area of a trapezium in order to do this.

So, to calculate the area of trapezium, remember, you add together the parallel sides. So, in my case that’s this 10-centimetres side here and then this side, which is six centimetres. So, I’m gonna add them together. And then, I halve that. What I’m essentially doing is finding an average of that pair of parallel sides. Then, I need to multiply it by the distance between the parallel sides. So, that’s this measure of seven centimetres. So, that’s the first part of the calculation. And all that’s doing is finding the area of that trapezium, the front face of the prism.

Then, to turn it into a volume, I need to multiply by the depth of the prism. So, I need to multiply by eight. So, this gives me the full calculation that I need. If I then go ahead and evaluate that, then it gives me a volume of 448 cubic centimetres for this particular prism.

So, as a reminder then, we’ve seen that a prism is a 3D shape with a constant cross section, which means that if you cut into it at any point you see exactly the same face. And in order to calculate the volume of a prism, we work out the area of this constant cross section and then multiply it by the height of the prism. Remember, we calculate the volume for any type of prism in the same way. We just have to think about how we calculate the area of the cross section each time depending what two-dimensional shape it is. Whether it’s perhaps a square, a rectangle, a cylinder, a trapezium, a parallelogram, or some other two-dimensional shape.