7.07 Surface area of pyramids

Remember that a pyramid can have any polygon as a base. We can have square bases, triangular bases or even hexagonal bases. Then connect every vertex of the base to an apex point above the base, and you have a pyramid.

If the apex is directly above (ie perpendicular to) the center of the base it is a right pyramid.

We can identify a pyramid by the shape of the base. Square and rectangular based pyramids, are the most common you will come across in mathematics, but also in the real world.

For example,

Here are a few pyramids with a triangular base, a rectangular base and a square base.

We can calculate the area of the base using the appropriate formula for its shape, since its dimensions will match the dimensions of the pyramid.

Finding the area of the triangular faces is not quite as straightforward. The base length of each of the triangular faces is the length of the base side that they are joined to, which is easy to find. But since the triangular faces are tilted to meet at the vertex, the height of each triangle corresponds to the slope length (or slant height) of that face, not the height of the pyramid.

When calculating the surface area (SA) of a pyramid we need to add up the areas of individual faces. Make sure not to miss any faces but also try to look for clever methods, like using the fact that 2 faces might have the same area.

For right pyramids, the concept below will help in finding the surface area of a pyramid.

\text{Surface area of a pyramid} = \text{Area of the base}+ \text{Area of the triangles}

As we saw with the applet above, there are two pairs of congruent triangles that we can get the area.

• The area of the one pair of triangles: 2\times \dfrac{l \times s}{2}=l\times s

• The area of the other pair of triangles: 2\times \dfrac{w \times s}{2}=w\times s

• The rectangular base: l \times w

Examples

Example 1

Find the surface area of the following rectangular pyramid.

Worked Solution

Create a strategy

Remember the surface area of a rectangular pyramid consists of two pairs of congruent triangles and a rectangle. Each slant height of the pyramid will correspond to one of the pairs of identical triangle faces.

Apply the idea

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle \text{Area of the base}+ \text{Area of the triangles}
	\displaystyle =	\displaystyle l\times w +l\times s+w\times s	where l and w are the length and width of the rectangle, and s are the slant heights of each triangle
	\displaystyle =	\displaystyle 8 \times 7+ 8\times 10+7\times 8	Substitute the values
	\displaystyle =	\displaystyle 192\text{ cm}^{2}	Evaluate

Watch question walkthrough

Example 2

A square pyramid has a surface area of 96\text{ ft}^{2}. Each triangular face has an area of 15\text{ ft}^2.

Find the side length of the base of the pyramid, correct to the nearest foot.

Worked Solution

Create a strategy

The area of the base can be found by subtracting the surface area by the area of the triangles. There are four congruent triangular faces in a square pyramid.

Once we have the area of the square base, we can solve for the side length of the square.

Apply the idea

There are four triangular faces in a square pyramid. The total area of the triangualar faces is given by:

4\times 15 \text{ft}^2= 60\text{ ft}^2

\displaystyle \text{Area of the base}	\displaystyle =	\displaystyle \text{Surface area} - \text{Area of the triangles}
	\displaystyle =	\displaystyle 96 - \, 60	Substitute the values
	\displaystyle =	\displaystyle 36	Evaluate

\text{Area of the square base}=s^2, where s is the side length of the square base.

\displaystyle {s^2}	\displaystyle =	\displaystyle s \times s	Rewrite in expanded form
\displaystyle s \times s	\displaystyle =	\displaystyle 36	Consider what number times itself is equal to 36
\displaystyle s	\displaystyle =	\displaystyle 6	Evaluate

The side length is 6 \text{ ft}.