Volume of a pyramid
Definition:
The number of cubic units that will exactly fill a pyramid.
Try this
Drag the orange dots to adjust the base size and height of the pyramid and note how the volume changes.
The volume enclosed by a
pyramid
is one third of the base area times the
perpendicular
height. As a formula:
Where:
b is the area of the base of the pyramid
h is its height.
The height must be measured as the vertical distance from the apex down to the base.
Recall that a pyramid can have a base that is any
polygon,
although it is usually a
square.
To calculate the area of a polygon see:
In the figure above, drag the orange dots to change the height and base size. The base is square, but you can make it a
rectangle
by unchecking 'Keep base square".
Oblique pyramids
Recall that an
oblique pyramid
is one that 'leans over'  where the apex is not over the base center point.
Drag the apex left and right above to see this. It turns out that the volume formula works just the same for these. You must however use the perpendicular height in the formula. This is the vertical red line in the figure above.
To illustrate this, check 'Freeze height'. As you drag the apex left and right, watch the volume calculation and note that the volume never changes.
Enclosing prism
Recall that the
volume of a prism
is its base area times its height.
If you compare this to the formula of the pyramid, you will see one is exactly a third of the other.
This means that the volume of a pyramid is exactly one third the volume of the prism with the same base and height.
Such a prism is called the "circumscribed prism" of the pyramid  the smallest prism that can contain the pyramid. In the
figure above, select "Show prism" to see this.
Similarity to a cone
Both the
cone and pyramid have the same way of calculating the volume  one third the area of the base times height.
In fact you can think of a cone as a pyramid with an infinite number of faces.
In Pyramid definition you can see this by increasing the number of base sides
to a large number and noting how it begins to look like a cone.
Things to try
 In the figure above, click "hide details".
 Drag the orange dots to set the base shape and height of the pyramid.
 Calculate the volume of the pyramid using the formula
 Click "show details" to check your answer.
While you are here..
... I have a small favor to ask. Over the years we have used advertising to support the site so it can remain free for everyone.
However, advertising revenue is falling and I have always hated the ads. So, would you go to Patreon and become a patron of the site?
When we reach the goal I will remove all advertising from the site.
It only takes a minute and any amount would be greatly appreciated.
Thank you for considering it! – John Page
Become a patron of the site at patreon.com/mathopenref
Related topics
(C) 2011 Copyright Math Open Reference. All rights reserved
