Video: Volumes of Pyramids

In this video, we will learn how to find volumes of triangular or quadrilateral pyramids and solve problems including real-life situations.

13:03

Video Transcript

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

We can start by reminding ourselves of what a pyramid is. A pyramid is a three-dimensional shape where the base is a polygon, for example, a triangle, a square, a pentagon, or so on. And all the other sides are triangles that meet at the apex or vertex of the pyramid.

We can recall that there are two special types of pyramids. The first type is a right pyramid, which is a pyramid whose apex lies above the centroid of the base. The second type is a regular pyramid, which is a right pyramid whose base is a regular polygon. That is, all the sides of the base are of equal length and all the pyramid’s lateral edges are of equal length. And now, let’s think about the volume of a pyramid.

The volume of a three-dimensional shape is how much space that shape would take up. If we imagine filling this pyramid with water and then tipping that into a container, how much water would there be? So, let’s imagine we have a prism that has the same length, width, and height as the pyramid. If we took our pyramid filled with water and tipped it into the prism, then the volume of the water in the prism would be a third the height of the prism. We can, therefore, say that, to find the volume of a pyramid, it is equal to a third times the area of the base times the height of the pyramid. And this is the formula that we use to find the volume of a pyramid.

In this diagram, the area of the base is a rectangle, so we would multiply the length by the width. But of course, the polygon at the base can be any shape. For example, if it was a triangle, we’d need to use the formula to find the area of the triangle to work out the area of the base. So, now, let’s look at some questions involving the volume of a pyramid. In this video, we’ll just be looking at those pyramids that have a triangular base or quadrilateral base.

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

So, here, we have a pyramid with a rectangular base. We’re going to use the formula that the volume of a pyramid is equal to a third times the area of the base times the height. To find the area of the base, as we know that this is a rectangle, we’re going to multiply the length by the width, giving us six multiplied by four. And that’s 24 square centimeters.

To find the volume of the pyramid then, using the formula, we have that it’s equal to a third multiplied by the area of the base, which is 24. Multiplied by the height of the pyramid, which is nine centimeters. We can then find a third of either 24 or nine. In this case, we can find a third of 24 as eight. And so, our calculation is eight multiplied by nine, which is 72. And the units here, as it’s a volume, would be cubic centimeters. As we’re asked to give our answer to the nearest hundredth, we can indicate this with an answer with hundredths as 72.00 cubic centimeters.

We’ll now take a look at a question involving a triangular-based pyramid.

Determine, to the nearest tenth, the volume of the given solid.

We can see that this solid has a triangular base and three other triangular faces, which means that this is a triangular-based pyramid. And so, to find the volume, we can recall that the formula for the volume of a pyramid is that it’s equal to a third times the area of the base times the height. As we have a triangular pyramid here, the area of the base could be any of the four triangular sides. To keep it simple, however, let’s use the one that’s currently highlighted in orange here.

And so, to find the area of the base, we can use the formula that the area of a triangle is equal to half times the base times the height. And therefore, on this triangle on the base, we have a base of 10 and a height of eight. So, we calculate a half times 10 times eight. We could then simplify this to the calculation five times eight, which will give us 40, with units of square centimeters.

Using this in the volume of a pyramid, we’ll have one-third multiplied by 40 and multiplied by the height of this pyramid, which would be five centimeters. Therefore, the volume is 200 over three, or 200 divided by three, which is 66.6 repeating cubic centimeters. And to round it to the nearest tenth then, as our digits would be six six six continuing, this means that when we check our second decimal digit, we would have a value that’s five or more. And our answer for volume will round up to 66.7 cubic centimeters.

Work out the volume of a square-based pyramid with a height of 12 inches and a base length of five inches.

Let’s begin this question by sketching a diagram of this pyramid. As we’re told that this pyramid has a base length of five inches, since this is a square, we know that both the length and the width will be the same. The height of 12 inches refers to the perpendicular height of the pyramid. We can recall that the volume of a pyramid is equal to a third multiplied by the area of the base multiplied by the height.

So, to find the area of the base, which is a square in this case, we know that that will be the length multiplied by the length. So, that’s five times five, giving us 25 square inches. So, to find the volume of the pyramid, we take the area of our base, which is 25. So, we have a third times 25 times the height, which is 12 inches. We can simplify this calculation by finding a third of 12, which is four. So, we now have the calculation 25 multiplied by four, which is 100. And our units here will be cubic inches. So, our final answer for the volume of the pyramid is 100 cubic inches.

In the following question, we’ll see an example of how sometimes we might need to use the Pythagorean theorem to help us get all the values that we need for the volume of a pyramid.

Find the volume of the following regular pyramid rounded to the nearest hundredth.

As we’re told that this is a regular pyramid, this means that the base of the pyramid is a regular polygon. And so, we know that the lengths of all the sides on the triangle at the base will be 14 centimeters. To find the volume of a pyramid, we calculate a third multiplied by the area of the base multiplied by the height.

Let’s begin by working out the area of this triangle at the base of the pyramid. To find the area of a triangle, we work out a half times the base times the height. If we take a closer look at this triangle, we know that it will have all three sides of 14 centimeters. Which means that we know the base length, but we don’t know the height of this triangle. As we have a right-angled triangle, we could apply the Pythagorean theorem.

The Pythagorean theorem tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides. So, let’s use the Pythagorean theorem to work out the height of our triangle. We can define the height here to be the unknown value of 𝑥. So, the hypotenuse here, 𝑐 squared, will be 14 squared, which is equal to 𝑥 squared, plus seven squared since seven is half of 14. And we’re using the smaller half of the original base triangle.

Evaluating the squares then, we have 196 equals 𝑥 squared plus 49. We then rearrange by subtracting 49 from both sides of the equation, giving us 147 equals 𝑥 squared. And we then take the square root of both sides to give us the square root of 147 equals 𝑥. We’re going to keep our answer in this square root form as we continue on through the question.

So, now, we’ve worked out that the height of the triangle is the square root of 147. We can work out the area of this triangle using the formula. So, our area is equal to half times the base, which is 14, and multiplied by the height, which is the square root of 147. We can then simplify this calculation to seven multiplied by the square root of 147. We could at this stage evaluate this using a calculator, but as we still need to work out the volume, we can keep it in this format of seven root 147.

Now we’ve worked out the area of the triangle — that’s the area of the base of the pyramid — we can now go ahead and work out the volume of the pyramid. So, our volume is equal to one-third multiplied by the area of the base, which is seven root 147, multiplied by the height of the pyramid, which is 17 centimeters. Using our calculator, we can evaluate this as 480.93277 and so on. And rounding to the nearest hundredth means that we check our third decimal digit to see if it is five or more. And as it is not, then our answer stays as 480.93. And our units here will be cubic centimeters.

In the final question, we’ll be given the volume of a pyramid and the height, and we’ll need to work out the perimeter of the polygon at the base of the pyramid.

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

So, let’s model our square pyramid with volume of 372 cubic centimeters. The height of 31 centimeters refers to the perpendicular height of the pyramid. We’re asked to work out the perimeter of the base of this square pyramid. That’s the distance all the way around the outside.

Let’s consider what we know about the volume of a pyramid. We can recall that the volume of a pyramid is equal to one-third multiplied by the area of the base multiplied by the height. This won’t directly help us to work out the perimeter. But if we could work out the area of the base, we could then go ahead and work out the perimeter. So, let’s start by filling in the information that we know into this formula.

This will give us 372 — that’s the volume of the pyramid — equals one-third times the area of the base times 31, which is the height of the pyramid. We can simplify the right-hand side by writing a third multiplied by 31 as 31 over three. We then want to isolate the area of the base, so we perform the inverse operation to multiplying by 31 over three. And that’s to divide by 31 over three, which is the same as multiplying by three over 31. So, we have 372 times three over 31 is equal to the area of the base.

We can then evaluate this without a calculator by noticing that 31 goes into 372 12 times. Meaning that the area of the base is equal to 12 times three, which is 36 square centimeters. So, now that we’ve worked out the area of the square on the base, we can use this to work out the length of the sides and, therefore, to calculate the perimeter.

So, if our side lengths are 𝑥 by 𝑥, this means that the area 𝑥 squared would be equal to 36. Therefore, the length 𝑥 is equal to the square root of 36, which is six centimeters. And so, the perimeter, which is the distance around the outside of this square, is equal to six plus six plus six plus six, which is 24 centimeters.

Now, let’s summarize what we’ve learned in this video. We recall that pyramids are three-dimensional geometric shapes where the base is a polygon and all other sides are triangles that meet at the apex. We also looked at the meaning of a right pyramid and a regular pyramid.

We saw that the volume of a pyramid is one-third of the volume of the prism of the same base and height. We wrote this as a formula that the volume of a pyramid equals a third multiplied by the area of the base multiplied by the height. And finally, as we saw in one of our examples, there may be times when we need to use the Pythagorean theorem to help us calculate unknown dimensions.

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