Volume of a Pyramid

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.
• 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.

Pyramids

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.

Volume Of Pyramids

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 to calculate the volume of square pyramids.
Worksheet to 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 cm³

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 Pythagoras’ theorem,

Area of triangle
=
= 108 cm²

Volume of pyramid

= 720 cm³

This video gives the formula of the volume of a pyramid

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.

Word problems about the volume of pyramids

Find the height of a pyramid given the volume and base dimensions.

Problem: 1) A square pyramid has a height of 7 m and a base that measures 2 m on each side. a) 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 in³. What is the volume of a pyramid with the same base and height?

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

Demonstrate the formula of the volume of a pyramid

This video will demonstrate that the volume of a pyramid is one-third that of a prism with the same base and height and that the volume of a cone is on-third that of a cylinder with the same base and height. This is not a formal proof.

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