Solid geometry is concerned with three-dimensional shapes. In this lesson, we will learn

- what is a pyramid?
- how to find the volume of a pyramid with rectangular or square bases.
- how to find the volume of a pyramid with different types of bases.
- how to solve word problems about pyramids.
- how to demonstrate the relationship between the volume of a pyramid and the volume of a prism with the same base and height.

Related Topics: Other Topics in Geometry

A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. The lateral faces meet at a common **vertex**. The **height** of the pyramid is the perpendicular distance from the base to the vertex.

A pyramid is a regular pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles. The pyramid is named after the shape of its base. A rectangular pyramid has a rectangle base. A triangular pyramid has a triangle base.

A **right pyramid** is a pyramid in which the vertex is vertically above the center of the base. If the vertex is not vertically above the center of the base then it is an oblique pyramid.

The volume of a pyramid is equal to one-third the product of the area of the base and the height.

The volume of a pyramid is given by the formula:

_{}

Worksheet calculate the volume of square pyramids | Worksheet calculate the volume of prisms and pyramids |

**Example: **

Find the volume of a pyramid with a rectangular base measuring 6 cm by 4 cm and height 10 cm.

Solution:

Volume

**= ** 80 cm^{3}

*Example: *

The following figure is a right pyramid with an isosceles triangle base. Find the volume of the pyramid if the height is 20 cm.

Solution:

First, we have to calculate the area of the base.

To do that, we would need to get the height of the isosceles triangle that forms the base.

Using Pythagorean theorem,

Area of triangle

=

= 108 cm^{2
}

Volume of pyramid

**= ** 720 cm^{3}

The following video shows how to find the Volume of a Pyramid.

The following video gives another example of how to calculate the volume of a pyramid.

Volume of a pyramid : Calculate height using Pythagorean Theorem

How to find the volume of a square pyramid and a triangular pyramid and compare how they are the same and how they are different

Volume of a Pentagonal Pyramid

Find the surface area of a regular pentagonal pyramid given an altitude of 4 and a slant height of 5.

Given the altitude and slant height we can find the apothem. Using the apothem, we can find the area of the base. The volume of the pyramid is 1/3 the area of the base multiply by the height.

Volume of a Hexagonal Pyramid

Find the volume of a hexagonal pyramid that has a height of 8 and a base edge of 10

The base of the following pyramid is a square. If the volume of the pyramid is 360 ft

Problem:

1) A square pyramid has a height of 7 m and a base that measures 2 m on each side. Find the volume of the pyramid. Explain whether doubling the height would double the volume of the pyramid.

2) The volume of a prism is 27 in^{3}. What is the volume of a pyramid with the same base and height?

The volume of a pyramid is 80cm^{3}, base is a triangle, find the height

Problem: Calculate the volume of a composite figure with a pyramid and a prism.

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