how to find the area of a pyramid

How do you calculate the volume and surface area of a pyramid?

Jenn (B.S., M.Ed.) of Calcworkshop® teaching pyramids

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

Great question!

That's exactly what we're going to do in today's geometry lesson.

You're going to learn how to determine areas and volumes of regular pyramids.

So let's get started!

Did you know that all the sides of a pyramid are triangular?

All pyramids are polyhedra with a single base and triangular lateral faces.

The altitude is the perpendicular segment from the vertex, or apex, to the base, as Math Is Fun states, and the height of the pyramid is the length of the altitude, whereas the slant height is the length of any lateral face.

pyramid properties

Pyramid Properties

Now calculating the volume, couldn't be easier. In fact, the volume of any pyramid is one-third the area of the base times the height.

volume of a pyramid formula

Volume of a Pyramid Formula

Finding the surface area of a pyramid is done by first finding the area of the base and the area of each lateral face. Thankfully, if we are given a regular pyramid, there are formulas that we can use to make our calculations easier.

First, a regular pyramid is a pyramid with a base that is a regular polygon. This means that all the sides are equal and all of the angles are equal.

Secondly, all lateral faces of a regular pyramid are congruent isosceles triangles.

Thirdly, the lateral area is the sum of the area of a regular pyramid's lateral faces and is found by taking half the product of the perimeter of the base and the slant height of one of the lateral faces.

calculate the lateral area, surface area, and volume of a pyramid

How to Calculate the Lateral Area, Surface Area, and Volume of a Pyramid

lateral area of a pyramid formula

Lateral Area of a Pyramid Formula

Lastly, the surface area of a pyramid is the sum of the area of the base and the lateral area!

surface area of a pyramid formula

Surface Area of a Pyramid Formula

This lesson will review several key concepts from previous lessons in including:

Right triangles
Equilateral triangles
Trigonometric ratios
SOH-CAH-TOA
Area of a regular polygon

In the video below you'll expand upon that knowledge and learn how to:

Find the volume of a pyramid.
Calculate the lateral and surface area for regular pyramids.
Determine the volumes for regular pyramids.

Video – Lesson & Examples

58 min

Introduction to video: area and volume of pyramids
00:00:16 – What is a pyramid? Formulas for finding volume, lateral area, and surface area
00:08:57 – Find the volume of each type of pyramid (Examples #1-3)
Exclusive Content for Member's Only

00:14:39 – Find the lateral area and surface area of a square pyramid (Example #4)
00:19:14 – Find the lateral and surface area of a triangular pyramid (Example #5)
00:26:35 – Find the surface and lateral area of a hexagonal pyramid (Example #6)
00:40:29 – For an octagonal pyramid, find the lateral area and surface area (Example #7)
00:51:58 – Find the volume and surface area of a composite solid (Example #8)
Practice Problems with Step-by-Step Solutions
Chapter Tests with Video Solutions