Surface area of a square pyramid

It is not complicated to derive the formula of the surface area of a square pyramid. Start with a square pyramid as shown below and call the length of the base s and the height of one triangle l.

l is the slant height. It is not for no reason this height is called slant height!

Take a close look again at the slant height (red line) and you can see that the line is not vertical.

Here is how to derive the surface area of a square pyramid.

Surface area of the square pyramid = area of the base + area of 4 triangles.

The area of the square base is s²

The area of one triangle is (s × l)/2

Since there are 4 triangles, the area is 4 × (s × l)/2 = 2 × s × l

Therefore, the surface area, call it SA is SA = s²  +  2 × s × l

A couple of examples showing how to find the surface area of a square pyramid.

Example #1:

Find the surface area of a square pyramid with a base length of 5 cm, and a slant height of 10 cm.

SA = s²  +   2 × s × l

SA = 5²   +   2 × 5 × 10

SA = 25   +   10 × 10

SA = 25  +   100

SA = 125 cm²


Example #2:

Find the surface area with a base length of 3 cm, and a slant height of 2 cm.

SA = s²  +  2 × s × l

SA = 3²  +   2 × 3 × 2

SA = 9  +   6 × 2

SA = 9  +   12

SA = 21 cm²


Example #3:

Find the surface area with a base length of 1/2 cm, and a slant height of 1/4 cm.

SA = s²  +   2 × s × l

SA = (1/2)²  +   2 × 1/2 × 1/4

SA = (1/2)×(1/2)  +   2 × 1/2 × 1/4

SA = 1/4  +  1 × 1/4

SA = 1/4 + 1/4

SA = 2/4

SA = 1/2 cm²