Surface Areas of Pyramid (right pyramid with a square base)

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We are going to explain how to find surface areas of a right pyramid with a square base. Most simply, it is nothing but areas of all four triangular faces and the square base at the bottom.
Thus, we can get surface areas (A) of the pyramid as,

A = 4 x (area of a triangular face) + (area of the square base)

If you want to know how to calculate the surface area in different manners, keep reading this article! This will explain more details in advance.

First, let's check what is a right pyramid with a square base. Have a look at Figure 1 below. 

Figure 1

We can see that there are four triangular faces and one square base for the above (Figure 1) pyramid. We can call it a right pyramid with a square base ONLY if the line segment connecting the apex and the midpoint of the square base is perpendicular to the base as shown in the above figure. The figure shows the names for each important part of the pyramid. Also, all the triangular faces have the exact same surface area in every right pyramid with a square base.

Now you should have a proper idea about the pyramid and its property. Then, let's move to find out the surface area of this pyramid. It is more useful and clear if you can see each different surface separately as below. Therefore, the first step is to separate them out.

There are four triangular faces and one square surface. Each different surface can be shown as in Figure 2 below. 

Figure 2

Thus, if we take the total surface area of the square-based right pyramid as A, it can be found as the summation of areas of all faces.

According to Figure 2 above,

Area of the square = a x a

Area of a triangular face = (1/2) x a x l

Since there are four triangular faces, we should multiply it by 4 to get areas of all 4 triangular faces.

Therefore, area of 4 triangular faces = 4 x (1/2) x a x l = 2al

According to that, A can be calculated as,

IMPORTANT - You have to carefully obtain the value "l" (right height of a triangular face) from the given values of the pyramid. This is not the length of the slant edge as shown in Figure 1. If you cannot grab it, look at Figure 3 below. 

Figure 3

Figure 3 shows the 'l' clearly. When finding the area of a triangular face,  get this 'l' using some other way if it is not given directly. The most typical way is the use of the Pythagoras' theorem when the right height of the pyramid and the side length of its base are given. Look at the following two examples which show the correct way of calculating.

Example 01;

Area of the square base = 10 x10 = 100

Area of a triangular face = 0.5 x 10 x 15 = 75

Areas of all 4 triangular faces = 75 x 4 = 300

thus,

Total surface areas of the pyramid = 100 + 300 = 400 cm²

Example 02;

Here, we first have to get the lenght of 'l'. We can apply Pythagoras' theorem for that as follows,

Thus, l = 10 cm

Area of the square base = 12 x 12 = 144

Area of a triangular face = 0.5 x 12 x 10 = 60

Areas of all 4 triangular faces = 60 x 4 = 240

thus, 

Total surface areas of the pyramid = 144+ 240 = 384 cm²

Hope this could help you to learn how to find surface areas of a right pyramid with a square base. If there is anything to be clarified, leave a comment below or contact us via the contact form on the right-hand side of our page.

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