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Volume of Right Pyramids


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.


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.

$\text{Volume of Cube }=s^3$Volume of 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.  

Volume of Pyramid

$\text{Volume of Pyramid }=\frac{1}{3}\times\text{Area of base }\times\text{Height }$Volume of Pyramid =13×Area of base ×Height

Worked Examples

question 1

Find the volume of the square pyramid shown.

A triangular pyramid is depicted with its faces visible and outlined. A vertical dashed line, representing the height, is drawn from the apex of the pyramid perpendicular to the base and is labeled as 6$cm$cm. One side of the base is labeled as 10$cm$cm. All the sides of the base are drawn with a single tick mark indicating that the measurements of the sides are all equal. 

question 2

A small square pyramid of height $4$4 cm was removed from the top of a large square pyramid of height $8$8 cm forming the solid shown. Find the exact volume of the solid.

A  square pyramid is depicted. The top portion of the pyramid is removed, indicated by dashed lines, creating a new top base, with its side measuring 4 cm. The bottom base of the square pyramid has a side measuring 8 cm. The vertical height of the pyramid is composed of two measurements, the removed top smaller pyramid measuring 4 cm, and the bottom part measuring 4 cm. 
  1. Give your answer in exact form.

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