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!

The word slant refers also to something that is oblique or bent, or something that is not vertical or straight up. Basically, anything that is not horizontal or vertical!

The area of the square 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 :

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  +  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  +  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 = 1/4  +  1 × 1/4

SA = 1/4 + 1/4

SA = 2/4

SA = 1/2 cm2

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