12.20 Volume of pyramids and cones

Volume of a pyramid

A pyramid is formed when the vertices of a polygon are projected up to a common point (called a vertex).  A right pyramid is formed when the apex is perpendicular to the midpoint of the base.

We want to be able to calculate the volume of a pyramid. Let's start by thinking about the square based pyramid.

Exploration

Think about a cube, with side length $s$s units.  Now lets divide the cube up into 6 simple pyramids by joining all the vertices to the midpoint of the cube.  

This creates $6$6 square based pyramids with the base equal to the face of one of the sides of the cube, and height, equal to half the length of the side.

$\text{Volume of the cube }=s^3$Volume of the cube =s3

$\text{Volume of one of the pyramids }=\frac{s^3}{6}$Volume of one of the pyramids =s36​

Now lets think about the rectangular prism, that is half the cube.  This rectangular prism has the same base as the pyramid and the same height as the pyramid.  

Now the volume of this rectangular prism is $l\times b\times h=s\times s\times\frac{s}{2}$l×b×h=s×s×s2​= $\frac{s^3}{2}$s32​.

We know that the volume of the pyramid is $\frac{s^3}{6}$s36​ and the volume of the prism with base equal to the base of the pyramid and height equal to the height of the pyramid  is $\frac{s^3}{2}$s32​.

$\frac{s^3}{6}$s36​	$=$=	$\frac{1}{3}\times\frac{s^3}{2}$13​×s32​	Breaking $\frac{s^3}{2}$s32​ into two factors
$\text{Volume of pyramid}$Volume of pyramid	$=$=	$\frac{1}{3}\times\text{Volume of rectangular prism}$13​×Volume of rectangular prism	Using what we found in the diagrams
	$=$=	$\frac{1}{3}\times\text{Area of base}\times\text{height }$13​×Area of base×height	Previously shown

So what we can see here is that the volume of the pyramid is $\frac{1}{3}$13​ of the volume of the prism with base and height of the pyramid.  

Of course this is just a simple example so we can get the idea of what is happening.  However, if we performed the same exercise with a rectangular prism we would actually get the same result (try it out!), so the volume for any pyramid is given by the formula:

 

Volume of a cone

The volume of a cone has the same relationship to a cylinder as we just saw that a pyramid has with a prism. As such, the volume of a cone can be found using the formula:

Due to its circular base and curved surface, the tools required to derive this formula lie outside the scope of this level of mathematics. Regardless, it suffices to know this formula and how to apply it when solving problems involving the volume of a cone.

 

Practice questions

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