# Surface Area of a Pyramid – Explanation & Examples

Before we get started, let’s review what a pyramid is. In geometry, a pyramid is a three-dimensional solid whose base is any polygon, and the lateral faces are triangles.

In a pyramid, the lateral faces (which are triangles) meet at a common point known as a vertex. The name of a pyramid is derived from the name of the polygon forming its base. For example, a square pyramid, a rectangular pyramid, a triangular pyramid, a pentagonal pyramid, etc.

The surface area of a pyramid is the sum of the area of the lateral faces.

This article will discuss how to find the total surface area and lateral surface area of a pyramid.

## How to Find the Surface Area of a Pyramid?

To find the surface area of a pyramid, you have to get the area of the base, then add the area of the lateral sides, which is one face times the number of sides.

### Surface of a pyramid formula

The general formula for the surface area of any pyramid (regular or irregular) is given as:

Surface area = Base area + Lateral area

Surface area = B + LSA

Where TSA = total surface area

B = base area

LSA = lateral surface area.

For a regular pyramid, the formula is given as:

The total surface area of regular pyramid = B + 1/2 ps

where p = perimeter of the base and s = slant height.

Note: Never confuse the slant height (s) and height (h) of a pyramid. The perpendicular distance from the vertex to the base of a pyramid is known as the height (h), while the diagonal distance from the pyramid’s apex to the edge of the base is known as the slant height (s).

#### Surface area of a square pyramid

For a square pyramid, the total surface area = b (b + 2s)

Where b = the base length and s = slant height

#### Surface area of a triangular pyramid

The surface area of a triangular pyramid = ½ b (a + 3s)

Where, a = apothem length of a pyramid

b = base length

s = slant height

#### Surface area of a pentagonal pyramid

The total surface area of a regular pentagonal pyramid is given by;

Surface area of a pentagonal pyramid = 5⁄2 b (a + s)

Where, a = apothem length of the base

and b = side length of the base, s = slant height of the pyramid

#### Surface area of the hexagonal pyramid

A hexagonal pyramid is a pyramid with a hexagon as the base.

The total surface area a hexagonal pyramid = 3b (a + s)

## Lateral Surface Area of a Pyramid

As stated earlier, the lateral surface area of a pyramid is the area of lateral faces of a pyramid. Since all the lateral faces of a pyramid are triangles, then the pyramid’s lateral surface area is half the product of the perimeter of the pyramid’s base and the slant height.

Lateral surface area (LSA = 1/2 ps)

Where p = perimeter of the base and s = slant height.

Let’s gain an insight into the surface area of a pyramid formula by solving a few example problems.

Example 1

What is the surface area of a square pyramid whose base length is 4 cm and slant height is 5 cm?

Solution

Given:

Base length, b = 4 cm

Slant height, s =5 cm

By the formula,

Total surface area of a square pyramid = b (b + 2s)

TSA = 4(4 + 2 x 5)

= 4(4 + 10)

= 4 x 14

=56 cm2

Example 2

What is the surface area of a square pyramid with a perpendicular height of 8 m and base length of 12 m?

Solution

Given;

Perpendicular height, h = 8 m

Base length, b =12

To get the slant height s, we apply the Pythagorean Theorem.

s = √ [82 + (12/2)2]

s = √ [82 + 62]

s = √ (64 + 36)

s =√100

= 10

Hence, the slant height of the pyramid is 10 m

Now, calculate the surface area of the pyramid.

SA = b (b + 2s)

= 12 (12 + 2 x 10)

= 12(12 + 20)

= 12 x 32

= 384 m2.

Example 3

Calculate the pyramid’s surface area, whose slant height is 10 ft, and its base is an equilateral triangle of side length, 8 ft.

Solution

Given:

Base length = 8 ft

Slant height = 10 ft

Apply the Pythagorean theorem to get the apothem length of the pyramid.

a = √ [82 – (8/2)2]

= √ (64 – 16)

= √48

a = 6.93 ft

Thus, the apothem length of the pyramid is 6.93 ft.

But, the surface area of a triangular pyramid = ½ b (a + 3s)

TSA = ½ x 8(6.93 + 3 x 10)

= 4 (6.93 + 30)

= 4 x 36.93

= 147.72 ft2

Example 4

Find the surface area of a pentagonal pyramid whose apothem length is 8 m, base length 6 m, and slant height is 20 m.

Solution

Given;

Apothem length, a = 8 m

Base length, b = 6 m

Slant height, s = 20 m

Surface area of a pentagonal pyramid = 5⁄2 b (a + s)

TSA = 5/2 x 6(8 + 20)

= 15 x 28

= 420 m2.

Example 5

Calculate the total surface area and lateral surface area of a hexagonal pyramid with the apothem as 20 m, base length as 18 m, and slant height as 35 m.

Solution

Given;

apothem, a = 20 m

Base length, b =18 m

Slant height, s = 35 m

The surface area a hexagonal pyramid = 3b (a + s)

= 3 x 18(20 + 35)

= 54 x 55

= 2,970 m2.

The lateral surface area of a pyramid =1/2 ps

Perimeter, p = 6 x 18

= 108 m

LSA = ½ x 108 x 35

= 1,890 m2

### Practice Questions

1. What is the surface area of a square pyramid with a base length of $12$ inches and a slant height of $13$ inches?

2. What is the surface area of a square pyramid with a perpendicular height of $15$ ft and base length of $40$ ft?

3. What is the pyramid’s surface area, with a slant height of $12$ m, and its base is an equilateral triangle with a length of $10$ m?

4. What is the area of the base of a square pyramid if it has a total surface area of $1440$ squared feet and a height of $24$ feet?

5. What is the total surface area of a hexagonal pyramid with the apothem as $24$ ft, base length as $15$ ft, and slant height as $25$ ft?