Volume of a Cone
Explained for the Curious Elementary Math Student

The formula for the Volume of a cone, is all your student NEEDS to know. However for some students this is simply not enough. Sometimes a student WANTS to know where it comes from. This page is for those students. Here, I will show, step by step how this formula came about.





The Volume of a Cone from First Principles

We must first take a look at the properties of Triangular Prisms.

Look at the image to the left. This is a Triangular prism with Base ACDF and two triangles, ABC and EFD. For our purposes here, I will refer to its base and B and it has a height of h.

This prism can be cut into three different pyramids that DO NOT OVERLAP. It can also be shown that these pyramids have the same volume. This tells us that each pyramid, within the Prism whose volume is it base multiplied by its height is Bh.

If the three pyramids are of equal volume, thenthe volume of each pyramid is:

Bh/3


This pyramid has base ABC and vertex E


This pyramid has base ACF and vertex E.

This pyramid has base FDC and vertex E.

Every pyramid, is EXACTLY one third the volume of a triangular prism with the same base and height - so...

the volume of ANY Pyramid is .... (Base x height) divided by three.

When detailing the surface area of the pyramid, I explained how, when the number of sides of a polygon subscribed in a circle continue to increase, the polygons perimeter gets closer and closer to the circumference of the circle.

We use the same theory here, to arrive at the conclusion that the volume of a cone is also:

Where B is the area of the base of the cone and h is its perpendicular height.

My explanation for the surface area of a cone may also be of interest to you.

As always, don't forget to use our free printable geometry worksheets to offer math practice to your child.

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