Lesson Video: Volumes of Pyramids Mathematics

In this video, we will learn how to find the volume of pyramids and to solve problems including real-life situations.

14:50

Video Transcript

In this video, we’ll learn how to find the volume of pyramids and how to solve problems including real-life situations. Let’s begin by defining what pyramids are and the different words we use for the parts of a pyramid along with the different types of pyramids.

Pyramids are three-dimensional geometric shapes where the base is a polygon and all the other sides are triangles that meet at the apex. We can have many different types of pyramids, but sometimes we can even reference these pyramids by naming them according to the polygon at the base, for example, a square pyramid, triangular pyramid, or even pentagonal or hexagonal pyramids. We have two other important definitions as well. A right pyramid is a pyramid whose apex lies above the centroid of the base. Then we have a more specific type of right pyramid and that’s a regular pyramid, and that’s a right pyramid where the base is a regular polygon. That means that all the sides of the base are equal length and all the pyramid’s lateral edges are of equal length.

Before we look at the volume of a pyramid, let’s make a distinction between the slant height and perpendicular height of a pyramid. The perpendicular height of the pyramid is the distance from the apex to the base. The slant height is the distance measured along a lateral face from the apex to the edge of the base. In other words, it is the height of the triangle, comprising one of the lateral faces. Now, let’s look at the volume of a pyramid of perpendicular height ℎ. Then, let’s imagine that we completely fill this pyramid with something like water. If we then poured this water in the pyramid into a prism of the same base and height, we would observe that the level of the water is exactly at one-third of the height of the prism. This allows us to define a general rule for any pyramid.

The volume of a pyramid is one-third of the volume of the prism of the same base and perpendicular height. We can calculate this by working out one-third multiplied by the area of the base multiplied by the perpendicular height. Let’s see how we can put this formula into practice in the first example.

Determine, to the nearest hundredth, the volume of the shown pyramid.

In order to answer this question, we can recall that to find the volume of a pyramid, we calculate one-third times the area of the base times the perpendicular height. In order to use this formula, although we are given the perpendicular height as nine meters, we’re not given the area of the base, so we’ll need to calculate it. Because the base is a triangle, we’ll use the formula to find the area of a triangle, which is a half times the base times the height. Notice that with these two formulas, we use the height twice, but it’s not the same height. For the area of a triangle, the height is the height of the triangle and for the height in the volume formula, that’s the height of the pyramid.

So, let’s work out the area of the triangle. So, we can take six as the base and 4.7 as the height, and we can use 4.7 directly because we’re given that this is a perpendicular height. When we work out this calculation, we get an answer of 14.1. Now, remember that this is an area and the units we’re given are in meters, so our area will be in meters squared. We can now use this value to calculate the volume of a pyramid, filling in the value that the area of the base is 14.1 and the height of the pyramid this time is nine meters. When we calculate one-third times 14.1 times nine, we get an answer of 42.3. And as this is a volume, we’ll have cubic units, so we’ll have a value of meters cubed. We can indicate the answer to the nearest hundredth by adding a zero in the hundredths column to give us the volume as 42.30 cubic meters.

In the next question, we’ll see how we can apply this volume formula to find an unknown height in a real-world context.

The Louvre Pyramid in Paris has a square base whose sides are 112 feet long. Given its volume is 296875 cubic feet, find the height of the pyramid to the nearest foot.

In this question, we’re given that there is a pyramid which has a square base, and these lengths of the square are 112 feet. We’re given the volume, but we need to calculate the height of this pyramid. We can remember that to find the volume of a pyramid, we calculate one-third times the area of the base times the perpendicular height. Here, we have got the volume of the pyramid. We need to calculate the perpendicular height. We don’t know the area of the base, but we can calculate this.

The area of a square is the length squared. So, in this case, we need to calculate 112 squared. This gives us 12544, and the units will be square feet. Now, we can apply the volume formula, filling in the given value 296875 as the volume and the area of the base that we’ve just calculated as 12544. Because we wish to find the value of ℎ, when we rearrange this, we can begin by multiplying both sides by three. Next, we can divide both sides by 12544. This gives 71.00 and so on is equal to ℎ. As we’re asked to give the value to the nearest foot, we can write the answer so that we find the height of the Louvre Pyramid as 71 feet.

We’ll now see how we can use a given volume of a pyramid and its height to find the perimeter of its base.

Given that a square pyramid has a volume of 372 cubic centimeters and a height of 31 centimeters, determine the perimeter of its base.

We haven’t been given a diagram here, but sometimes it can be helpful to sketch one. Firstly, we’re told that this is a square pyramid, so we’ll know that the length and the width will be exactly the same measurement on the base. We’re given the volume of the pyramid as 372 cubic centimeters and we’re given the perpendicular height as 31 centimeters. We can recall that the volume of a pyramid is equal to one-third times the area of the base times the perpendicular height. We can use the fact that we are given the volume of the pyramid and its perpendicular height to find the area of the base. This will allow us to then calculate the perimeter of the base.

We can substitute in the values then. The volume is 372 and the height is 31 to give us 372 is equal to one-third times the area of the base times 31. Multiplying both sides by three, we have 1116 is equal to the area of the base times 31. Next, we divide both sides by 31 to give us the area of the base is 36 square centimeters. So, how do we go from knowing the area of the base of this pyramid to finding its perimeter? Well, remember that the base is a square. So, if we define the length of one side as 𝑙, then all the other sides would be of length 𝑙. The area of a square would be calculated as 𝑙 squared. In this case, 𝑙 squared must have given us 36. To calculate 𝑙, we would take the square root of both sides, and the square root of 36 is six. And of course, the length units will be in centimeters.

To find the perimeter, that’s the distance around the outside edge. We could add six and six and six and six or more simply work out six times four, which would give us a value of 24. And because perimeter is still a length, then we’d have the units of centimeters. And so, we found that the perimeter of the base of this pyramid is 24 centimeters, and we did that using the volume by first calculating the area of the base.

So far in this video, we have seen examples of pyramids that have triangular or quadrilateral bases, of which their volume is easier to find using geometric techniques. In fact, if we have a more complex pyramid, we can still find the volume if we’re given the height and the area of its base. However, it can be more difficult to find the volume of a pyramid which has a base of five sides or more if we’re not given its area. But if we have a regular 𝑛-sided polygon at the base, we can use the following formula to help find its area. The area of a regular 𝑛-sided polygon of side length 𝑥 is given by the area of a polygon is equal to 𝑛𝑥 squared over four times the cot of 180 degrees over 𝑛. It can be useful to note down this formula along with the notes on the volumes of pyramids as it is used quite frequently. We’ll now see how this formula can be applied in the final example.

A regular pentagonal pyramid has base side length 41 centimeters and height 71 centimeters. Compute, to one decimal place, the volume of the pyramid.

Let’s start by sketching the pyramid with the given dimensions. So, here, we have the pyramid drawn. Because we’re told that it’s a pentagonal pyramid, that means the base will have five sides. And because we’re told that it’s a regular pyramid, we know that all the lengths on the base will be the same at 41 centimeters. We’re given that the height is 71 centimeters and that means that it’s the perpendicular height. Remember that the slant height would be the height of one of the triangles that makes up the lateral sides.

We can remember that to calculate the volume of a pyramid, we work out one-third times the area of the base times the perpendicular height. We’re not given the area of the base of this pyramid, but we can calculate it using the information that this is a regular pentagonal pyramid. And so, the base of this pyramid will be a five-sided polygon with all the sides the same length. The area of a regular 𝑛-sided polygon of side length 𝑥 is given by the area of a polygon is equal to 𝑛𝑥 squared over four times the cot of 180 degrees over 𝑛. Therefore, substituting in the values that the number of sides 𝑛 is equal to five and the side length 𝑥 is 41 gives us that the area of the polygon is equal to five times 41 squared over four times the cot of 180 degrees over five. This simplifies to 8405 over four times the cot of 36 degrees.

Since multiplying by the cotangent is equal to dividing by the tangent, we can write this as 8405 divided by four times the tan of 36 degrees. Using a calculator, we can find the decimal equivalent as 2892.122 and so on square centimeters. Because we haven’t quite finished with this value, we’re not going to round it yet. And when we use it in the next part of the calculation, we can keep the long decimal or alternatively use this fractional value in the step before.

Now we can work out the volume of the pyramid, remembering that the area of the base is the area of the polygon that we’ve just calculated. So, we can write that the volume of the pyramid is equal to one-third times 2892.122 and so on multiplied by the perpendicular height, which is 71. This gives us a value of 68446.899 and so on. And because it’s a volume, we’ll be working in cubic units, which means that the units will be cubic centimeters. We’re asked to round our answer to one decimal place, so we can give the volume of the pyramid here as 68446.9 cubic centimeters.

We can now summarize the key points of this video. We began with the definition that pyramids are three-dimensional geometric shapes where the base is a polygon, and all the other sides are triangles that meet at the apex. Next, we saw the two definitions. A right pyramid is a pyramid whose apex lies above the centroid of the base, and a regular pyramid is a right pyramid whose base is a regular polygon. That means that all the sides of the base are of equal length and all the pyramids’ lateral edges are of equal length.

Then, we saw the all-important rule. The volume of a pyramid is one-third of the volume of the prism of the same base and height. And so, the volume of a pyramid is equal to one-third times the area of the base times the perpendicular height. And finally, as we saw in the last example, in order to find the area of the base of a regular pyramid, we may need to apply the formula to find the area of an 𝑛-sided polygon of length 𝑥, which is given by the area of a polygon is equal to 𝑛𝑥 squared over four times the cot of 180 degrees over 𝑛.

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