# Pyramid in Math: Definition & Practice Problems

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• 0:05 Types of Pyramids
• 1:09 Pyramid Formulas
• 2:36 Example Problems
• 4:19 Lesson Summary

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Lesson Transcript
Instructor: Jennifer Beddoe
A pyramid is a 3-dimensional geometric shape formed by connecting all the corners of a polygon to a central apex. This lesson will define what a pyramid is and discuss the different types of pyramids and formulas surrounding them.

## Types of Pyramids

A pyramid is a 3-dimensional shape whose base is a polygon. Each corner of a polygon is attached to a singular apex, which gives the pyramid its distinctive shape. Each base edge and the apex form a triangle.

There are many types of pyramids. Most often, they are named after the type of base they have. Let's look at some common types of pyramids below.

Triangular pyramid has a triangle as its base:

Square pyramid has a square as its base:

Pentagonal pyramid has a pentagon as its base:

This list could go on and on (hexagonal pyramid, heptagonal pyramid, etc). There are also a few special names for pyramids that you should know.

Right pyramid - the apex of the pyramid is directly above the center of its base:

Oblique pyramid - the apex of the pyramid is not directly above the center of its base:

Regular pyramid - the base of this pyramid is a regular polygon:

Irregular pyramid - this type of pyramid has an irregular polygon as its base:

## Pyramid Formulas

There are formulas that can be used to find both the surface area and the volume of a pyramid. The surface area of a pyramid is the total area of all the surfaces that the pyramid has. To that end, the formula for finding the surface area when all of the side faces are the same is:

SA = (base area) + (1/2) * (perimeter) * (slant height)

The base area is the area of the base and can be determined based on what figure the base is. For example, if the base of the pyramid is a square, the formula for finding the area is:

A = s^2

The perimeter is the distance around the base of the pyramid. The slant height is the diagonal height from the center of one of the base edges to the apex.

If the pyramid has side faces that differ from each other (like in the case of an irregular pyramid), then the surface area equation is:

SA = (base height) + (lateral area)

In this case, you must take each side of the pyramid separately (including the base), find the areas, and then just add them together.

The volume of a pyramid can be found using this formula:

V = (1/3) * (base area) * (height)

The base area is, again, just the area of the base of the pyramid. However, in this case, the height is the length of a line from the apex that makes a right angle with the base.

## Example Problems

Let's take a look at some examples.

#### Example 1

A square pyramid has a height of 9 meters. If the side of the base measures 4 meters, what is the volume of the pyramid?

Since the base is a square, area of the base = 4 * 4 = 16 m^2.

Volume of the pyramid = (1/3) * (base area) * (height)

V = (1/3) * (16) * (9) = 48 m^3

#### Example 2

What is the surface area of the pyramid discussed in the first example?

In order to find the surface area, we must first find the slant height of the pyramid. Since we know the height and the base length, we can use the Pythagorean Theorem to find the slant length.

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