Volume & Surface Area: Surface Area of Pyramid & Cone

Q: How do you calculate the surface area of a right square pyramid?

The total surface area is: $A = s^2 + 2 s l$, where $s$ is the side length of the square base and $l$ is the slant height. If perpendicular height $h$ is given, calculate $l = \sqrt{h^2 + (s/2)^2}$ first.

Q: How do you calculate the surface area of a cone?

The total surface area is: $A = \pi r^2 + \pi r l$, where $r$ is the base radius and $l$ is the slant height. If height $h$ is given, calculate $l = \sqrt{h^2 + r^2}$ first.

Learn how to calculate the surface area of right pyramids, cones, and composite shapes. Interactive practice problems with step-by-step solutions.

Practice

📖 Learning Guide

1. Surface Area of a Right Pyramid

A right square pyramid has a flat square base and four triangular side faces. The surface area is the sum of the base area

B

and the lateral area

L

Base Area ( $B$ ): For a square base of side $s$ , $B = s^2$ .
Lateral Area ( $L$ ): Composed of four identical triangles with base $s$ and slant height $l$ . The total lateral area is $L = 4 \times \left(\frac{1}{2} s l\right) = 2 s l$ .
Slant Height ( $l$ ): If the perpendicular height $h$ is given instead of the slant height, calculate the slant height using the Pythagorean theorem on half the base side: $l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}$ .

Surface Area of a Pyramid (Square)

A = s^2 + 2 s l

Where

s

is the base side length and

l

is the slant height.

Slant Height of a Pyramid

l = \sqrt{h^2 + \left(\frac{s}{2}\right)^2}

Where

h

is the perpendicular height and

s

is the base side length.

2. Surface Area of a Cone

A right circular cone consists of a flat circular base and a curved lateral surface. Its surface area is the sum of the base area

B

and the lateral area

L

Base Area ( $B$ ): For a circle of radius $r$ , $B = \pi r^2$ .
Lateral Area ( $L$ ): The unwrapped curved side forms a circular sector with area $L = \pi r l$ , where $l$ is the slant height.
Slant Height ( $l$ ): If height $h$ is given, find the slant height using the Pythagorean theorem: $l = \sqrt{h^2 + r^2}$ .