# Surface Area of a Square Pyramid Formula     ## What is the Area of a Square Pyramid?

Let’s know about the surface area of a square-based pyramid. A square pyramid is also known as a polyhedron because it has a square base and four triangular faces. The apex is the point at which all four faces come together. The pyramid's faces connect the base to the apex. The surface area of the square pyramid formula is the sum of the surface areas of the base and the four triangular faces.

A rectangular pyramid looks like this:

The formula for the surface area of a square pyramid is simple to calculate. Begin with a square pyramid, as shown below, and denote the length of the base s and the Slant height of a square pyramid (l).

The slant height is denoted by l. This height is called slant height for nothing!

The term slant also refers to something that is oblique, bent, or not vertical or straight up. In other words, anything that isn't horizontal or vertical!

If you look closely at the slant height (red line), you will notice that it is not vertical.

Here is a formula for calculating the surface area of a square pyramid.

The surface area of a square pyramid is equal to the area of the base plus the area of four triangles.

The area of the square base is equal to s$^{2}$

The area of one triangle is equal to (s × l)/2

The slant height of a square pyramid equals to (l)

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

Therefore, the surface area, call it SA is Surface Area (SA)=  s$^{2}$  +  2 × s × l

### Lateral Surface Area of a Square Pyramid Formula

Consider a square pyramid with a base length (square side length) of 'a' and a height of 'l' for each side face (triangle) (this is also known as the slant height). Specifically, the base and height of each of the four triangular faces are 'a' and 'l'. As a result, the area of each such triangular face is 1/2 an l. So the total area of all four triangular faces is 4 (1/2 al) = 2 al. As a result, the lateral area of a square pyramid equals 2al.

What if we were given the pyramid height instead of the slant height? Assume that the pyramid's height (altitude) is 'h'. Then, using Pythagoras' theorem, l=[(a$^{2}$/4) + h$^{2}$]

By substituting this in the preceding formula,

The lateral area of a square pyramid equals 2al = 2a√[(a$^{2}$/4) + h$^{2}$]

A couple of examples that are given below show how to find the surface area of a square pyramid.

### Questions to be Solved

1) Find the Surface Area of a Square Pyramid with a Base Length of 5 Cm, and a Slant Height of 10 Cm.

Solution) Square pyramid formula = s$^{2}$ + 2 × s × l

Slant height of a square pyramid (l) = 10 cm

SA = 5$^{2}$ + 2 × 5 × 10

SA = 25 + 10 × 10

SA = 25 + 100

SA = 125 cm$^{2}$

2) Find the Surface Area with a Base Length of 3 Cm, and a Slant Height of 2 Cm.

Solution) Square pyramid formula = s$^{2}$ + 2 × s × l

Slant height of a square pyramid (l) = 2 cm

SA equals 3$^{2}$ + 2 × 3 × 2

SA = 9 + 6 × 2

SA = 9 + 12

SA = 21 cm$^{2}$

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