Surface Area of a Pyramid Formula
The surface area of a pyramid is the total area of all its faces, which includes the area of the base and the lateral faces (triangular faces connecting the base to the apex). Calculating the surface area depends on the shape of the pyramid’s base.
Here are the formulas for calculating the surface area of a pyramid, with detailed explanations and examples for each type of base shape.
Formulas for the Surface Area of a Pyramid
General Formula Using Base Area and Perimeter:
If the area \( A \) of the base and the perimeter \( P \) of the base are known, along with the slant height \( s \) of the pyramid,
The total surface area \( S \) can be calculated with the formula:
\( S = A + \dfrac{1}{2} P \times s \)
In this formula:
- \( A \) is the area of the base of the pyramid
- \( P \) is the perimeter of the base of the pyramid
- \( s \) is the slant height, or the distance from the apex to the midpoint of an edge of the base
Below are specific formulas for pyramids with different base shapes, with examples for each.
Surface Area of a Rectangular Pyramid
If the base of the pyramid is a rectangle with length \( l \) and width \( w \), the area of the base \( A \) is \( A = l \times w \), and the perimeter \( P \) of the base is \( P = 2(l + w) \). The total surface area \( S \) of the rectangular pyramid is then calculated as:
\( S = l \times w + \dfrac{1}{2} \times 2(l + w) \times s \)
Example 1: Calculating Surface Area of a Rectangular Pyramid
Problem: Find the surface area of a rectangular pyramid with length \( l = 5 \, \text{cm} \), width \( w = 3 \, \text{cm} \), and slant height \( s = 7 \, \text{cm} \).
Solution:
- Write down the formula: \( S = l \times w + \dfrac{1}{2} \times 2(l + w) \times s \).
- Substitute \( l = 5 \), \( w = 3 \), and \( s = 7 \): \( S = 5 \times 3 + \dfrac{1}{2} \times 2(5 + 3) \times 7 \).
- Calculate \( l \times w \): \( S = 15 + \dfrac{1}{2} \times 2 \times 8 \times 7 \).
- Multiply: \( S = 15 + 56 = 71 \, \text{cm}^2 \).
The surface area of the rectangular pyramid is \( 71 \, \text{cm}^2 \).
Surface Area of a Triangular Pyramid
If the base of the pyramid is a triangle with base \( b \), height \( h_b \), and side lengths \( a \), \( b \), and \( c \), the area of the base \( A \) is \( A = \dfrac{1}{2} b \times h_b \), and the perimeter \( P \) of the base is \( P = a + b + c \). The total surface area \( S \) of the triangular pyramid is then calculated as:
\( S = \dfrac{1}{2} b \times h_b + \dfrac{1}{2} P \times s \)
Example 2: Calculating Surface Area of a Triangular Pyramid
Problem: A triangular pyramid has a base with side lengths \( a = 4 \, \text{cm} \), \( b = 6 \, \text{cm} \), and \( c = 5 \, \text{cm} \), base height \( h_b = 4 \, \text{cm} \), and slant height \( s = 8 \, \text{cm} \). Find its surface area.
Solution:
- Calculate the area of the triangular base: \( A = \dfrac{1}{2} \times 6 \times 4 = 12 \, \text{cm}^2 \).
- Calculate the perimeter of the base: \( P = a + b + c = 4 + 6 + 5 = 15 \, \text{cm} \).
- Now use the total surface area formula: \( S = A + \dfrac{1}{2} P \times s \).
- Substitute \( A = 12 \), \( P = 15 \), and \( s = 8 \): \( S = 12 + \dfrac{1}{2} \times 15 \times 8 \).
- Multiply: \( S = 12 + 60 = 72 \, \text{cm}^2 \).
The surface area of the triangular pyramid is \( 72 \, \text{cm}^2 \).
Surface Area of a Pyramid with a Polygonal Base
For pyramids with polygonal bases (such as pentagonal or hexagonal pyramids), the area of the base \( A \) can be calculated using the appropriate formula for that polygon’s area, and the perimeter \( P \) as the sum of the polygon’s side lengths. Once \( A \) and \( P \) are known, the total surface area \( S \) is calculated as:
\( S = A + \dfrac{1}{2} P \times s \)