How to Calculate the Surface Area of Pyramids and Cones

Step-by-step Guide: Surface Area of Pyramids and Cones

Pyramid:

The surface area  of a pyramid can be found using the formula: Surface Area \(SA = \text{Base Area} + \frac{1}{2} \times \text{Perimeter of Base} \times \text{Slant Height}\)

Cone:

The surface area  of a cone can be found using the formula: Surface Area \(SA = \pi r^2 + \pi r \times \text{Slant Height}\)

Where:
\(r =\) radius of the base

Examples

Example 1:
Find the surface area of a square pyramid with a base side of \(5 \text{ cm}\) and a slant height of \(7 \text{ cm}\).

Solution:
Base Area \(= 5 \text{ cm} \times 5 \text{ cm} = 25 \text{ cm}^2\)

Perimeter of Base \(= 4 \times 5 \text{ cm} = 20 \text{ cm}\)

\( SA = 25 \text{ cm}^2 + \frac{1}{2} \times 20 \text{ cm} \times 7 \text{ cm} = 95 \text{ cm}^2 \)

Example 2:
Calculate the surface area of a cone with a base radius of \(6 \text{ cm}\) and a slant height of \(10 \text{ cm}\).

Solution:
\( SA = \pi (6 \text{ cm})^2 + \pi (6 \text{ cm})(10 \text{ cm}) = 301.44 \text{ cm}^2 \)

Practice Questions:

1. Determine the surface area of a triangular pyramid with a triangular base of area \(15 \text{ cm}^2\), base perimeter of \(18 \text{ cm}\), and a slant height of \(9 \text{ cm}\).
2. What is the surface area of a cone with a radius of \(4 \text{ cm}\) and a slant height of \(8 \text{ cm}\)?

Answers:

1. \( 96 \text{ cm}^2 \)
2. \( 150.8 \text{ cm}^2 \)