Video: Surface Areas of Pyramids

In this video, we will learn how to calculate the lateral and total surface areas of pyramids using their formulas.

14:25

Video Transcript

In this video, we will learn how to calculate the lateral and total surface areas of a pyramid. We will recall what a pyramid is, how to find its surface area, and then look at how to solve some typical problems involving these.

Let’s begin by thinking about what a pyramid actually is. Pyramids are three-dimensional shapes, where the base is a polygon, for example, a triangle, a square, or even a pentagon. And all the other sides are triangles that meet at the apex. In our first diagram, we have a square-base pyramid. In the second, we have a triangular-base pyramid. And in the third diagram, we have a pyramid where the base is a pentagon.

In this video, we’ll be focusing on square-base pyramids, rectangular-base pyramids, and triangular pyramids. We’ll also look at two specific types of pyramids, right pyramids and regular pyramids. A right pyramid is a pyramid whose apex lies above the centroid of the base. A regular pyramid is a right pyramid whose base is a regular polygon. Next, let’s recap what surface area is. The surface area is the total area of a three-dimensional shape. For example, if we wanted to find the surface area of this cube, we could more easily do this by visualizing the net of the cube.

So to find the surface area of the cube, we would find the area of the six individual faces and add them together. When we find the surface area of a three-dimensional shape, the units will be square units. It’s important to remember that surface area is different to volume even though they both involve three-dimensional shapes. When we’re calculating a volume, we’re working out how much space an object takes up. When we’re working out surface area, we’re working out the total area of each individual face.

So now let’s look at the main objective of this video. And that’s how to find the surface area of a pyramid. Here we have a typical square-base pyramid. To find the surface area, we’ll need to work out the area of the square on the base and then the area of the triangle at the back. And the two triangles at the sides and then the area of the triangle at the front and then add them together. In fact, this diagram is getting a little bit messy and hard to work with. So sometimes, drawing the net can make things simpler.

Once we’ve drawn the net, we can see that there’s a square and four triangles. We could then transfer over any information we’ve been given about the dimensions of the pyramid. We must pay close attention to the height of the pyramid. When we’re working with the surface area of a pyramid, one of the things we might be asked for is the lateral surface area. This is the total area of just the lateral sides, but excluding the base. In our square-base pyramid, that would just be the four lateral triangles, but not the area of the square on the base.

When we include the area of the base, that’s referred to as the total surface area. So to find the total surface area of this square-base pyramid, we’d have the total areas of the four triangles plus the area of the square on the base. When we’re looking at questions involving the surface area of a pyramid, we need to check if the word lateral is there or not.

So now let’s have a look at a few example questions.

If the given figure was folded into a square pyramid, determine its lateral surface area.

Here we have the net of a square pyramid. If we folded this net into a pyramid, we’d have a square on the base and four triangular lateral sides. Here, we’re asked to find the lateral surface area. So that means we’re going to find the area of the four triangles and add them together. We’re not concerned with the area of the square on the base of this pyramid. As we’re told that this is a square, we know that all four lengths on the base are congruent. And we’ll also have four congruent triangles.

To find the area of one of these triangles, we’re told that the base length is 14 centimeters and the height is given as 15 centimeters. To find the area of a triangle, we calculate a half multiplied by the base multiplied by the height. So for one of our triangles, with a base of 14 and a height of 15, we’ll have a half multiplied by 14 multiplied by 15. This can be simplified to seven times 15, giving us 105. And the units here will be in square centimeters since we’re dealing with an area.

So now we’ve found the area of one triangle, we can find the lateral surface area by adding together the area of the four triangles. As these triangles are congruent, this means we’ll be calculating four times 105, which is 420 square centimeters. So that’s our lateral surface area, remembering that we didn’t need to calculate the area of the square on the base.

Let’s now have a look at another question involving a square pyramid.

Determine the surface area of the given square pyramid, given that all of its triangular faces are congruent.

In this square pyramid, we can see that there’s a square on the base of 37 inches by 37 inches. And we can also see that there are four triangular lateral sides. We’re given that the height of the triangle is 44 inches. And we’re told that these four triangles are all congruent. We’re asked to calculate the surface area of this pyramid. And we can see that the drawn net here will help make things a little bit easier. To find the surface area of a pyramid, we find the area of all the lateral sides and add it to the area of the base. So, in this case, we’ll need to find the area of the four triangles and add it to the area of the square.

Let’s start by finding the area of one of these triangles. And we can recall that the area of a triangle is equal to half multiplied by the base multiplied by the height. For our triangle then, we have a base of 37 and a height of 44. So we’re working out a half multiplied by 37 multiplied by 44. We can simplify this calculation to 37 times 22, which gives us an answer of 814 square inches. In a moment, we can multiply that by four to find the area of four triangles. But let’s move on and find the area of a square.

We can recall that the area of a square is equal to the length multiplied by the length, which is 37 times 37. And so we have that the area of the square is 1369 square inches. Notice that even though we’re working with three-dimensional shapes, our units here will be square units since we’re still working with an area. A cubic unit would indicate a volume. Finally, to find our total surface area, we have the area of four triangles, which is four multiplied by 814, plus the area of our square, which is 1369. We can calculate this as 3256 plus 1369, giving us a final answer of 4625 square inches for the surface area of the pyramid.

In the next question, we’ll see an example where all of the dimensions that we need aren’t given to us, and so we need to use the Pythagorean theorem.

Find the total surface area of the given net, to the nearest hundredth.

We can see the net here that’s formed of a square and four triangles. When formed into a three-dimensional shape, we’d have a square pyramid. To find the total surface area of the net or the pyramid, we need to find the area of the four triangles and the area of the square and then add them together. Let’s start by finding the area of one of the triangles. If we look at the base of this triangle, we can see from the markings that it’s congruent with the sides of the square, meaning that this will also be two centimeters. We aren’t given the height of the triangle, but we are told that one of the other sides is 3.1 centimeters.

Modeling this as a right-angle triangle, we can take the unknown height to be 𝑥 and use the Pythagorean theorem. The Pythagorean theorem says that the square of the hypotenuse is equal to the sum of the squares on the other two sides. So in this triangle, we have a hypotenuse of 3.1, an unknown length of 𝑥, and the base is equal to half of two, which is one. So we’ll have 3.1 squared equals 𝑥 squared plus one squared. Solving this to find 𝑥, we can subtract one squared from both sides, giving us 3.1 squared minus one squared equals 𝑥 squared.

We can then evaluate the squares. And 3.1 squared is the same as 3.1 times 3.1. So we have 9.61 subtract one equals 𝑥 squared. Therefore, 8.61 equals 𝑥 squared. And taking the square root of both sides will give us that 𝑥 equals the square root of 8.61. As we’re going to continue using this value of 𝑥 in the next calculation, we keep it in the square root form rather than rounding to a decimal. Returning to the triangle since we found that the perpendicular height is the square root of 8.61, we can find its area using the formula, the area of a triangle is equal to a half multiplied by the base multiplied by the height.

We can then take the formula and plug in our values that the base is equal to two and the height is equal to the square root of 8.61. Since we can cancel a half multiplied by two as one, then we have that the area of the triangle is equal to root 8.61. And the units here will be square centimeters. Next, we calculate the area of the square at the base of the pyramid. And since we multiply the length by the length, we’ll have two multiplied by two, which is four square centimeters.

To find the total surface area then, we take four times the triangle area and add it to the area of the square, giving us four times the square root of 8.61 plus four. Using our calculator, we can evaluate this as 15.73712 and so on square centimeters. And to round our answer to the nearest hundredth means that we check our third decimal digit to see if it is five or more. And as it is, our answer will round up to 15.74 square centimeters. And this is our final answer for the total surface area of the net.

In the final question, we’ll see an example where we’re given the lateral surface area and asked to work out the total surface area.

A square pyramid has a lateral surface area of 42 square yards. If its slant height is three yards, determine its total surface area.

Let’s start by sketching out the square pyramid and filling in the relevant information. We’re told that this square pyramid has a lateral surface area of 42 square yards. The lateral surface area of a pyramid is the total area of the lateral sides or the triangles, but not including the area of the base. So the lateral surface area of this square pyramid will be the area of the triangle at the back plus the area of the two triangles at the sides plus the area of the triangle at the front. As we’re told that the lateral surface area is equal to 42 square yards, this is equivalent to saying that four times the area of one of these triangles is equal to 42. And therefore, the area of one triangle is equal to 42 divided by four or 10.5 square yards.

We can then take this information about the area of the triangle and combine it with the perpendicular height in order to work out the base length of this triangle. Knowing this would then allow us to calculate the area of the square on the base of this pyramid. We can recall that the area of a triangle is equal to a half multiplied by the base multiplied by the height. For our triangle then, we have an area of 10.5 and an unknown base length of 𝑏 and a height of three. Simplifying, we have that 10.5 is equal to three-halves 𝑏. In the next step of rearranging, we divide by three-halves, which is equivalent to multiplying by two-thirds. And so we’ve found that the base length 𝑏 of the triangle is equal to seven yards.

As the base of the triangle is also equivalent to the length of the square, we know that all the lengths on the square will be seven yards. To find the area of a square, we multiply the length by the length, which is seven times seven, giving us a value of 49 square yards. Finally, then, to find the total surface area, we were told that the lateral surface area is 42 square yards. That’s the area of all four triangles at the top of the pyramid. Therefore, for the total surface area, we must add on the area of the square to this lateral surface area, giving us a value of 91 square yards for the total surface area.

Now, let’s summarize what we learned in this video. 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 saw that there’s a difference between the lateral surface area and the total surface area of a pyramid. The lateral surface area is just the area of the lateral sides and excludes the base whereas the total surface area is the area of all sides including the base. We also saw that surface area is measured in square units, for example, square centimeters, square inches, or square yards. And finally, we saw that it can be helpful to draw a net of the pyramid. This allows us to visualize each face clearly so that we can work out the individual areas before adding them together.

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