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Working out the volume of a pyramid is fairly straightforward when you have the appropriate information.

The most common type of pyramid that is usually encountered, are pyramids with a square base.

Some examples of using this formula are further down the page, but we'll also give a run through of where this formula comes from first.

If we first consider a standard cube, where all sides are the same length.

The volume of this cube is given by **a** × **a** × **a**, or **a**^{3}.

Inside this cube, we can fit **6** pyramids with a square base, and apex/top at the center.

All pyramids would be the same size.

The image below shows one of the **6** pyramids inside the cube.

As the volume of the cube is given by **a**^{3},

and as the pyramid is one of **6** equal sized pyramids that can fit inside the cube.

The volume of this individual pyramid can be given by \bf{\frac{a^3}{6}}.

The pyramid has a height **h**, which is half the length of a cube side **a**.

As the top of the pyramid is the center of the cube.

So with **h** = \bf{\frac{a}{2}},
it's the case that **2h** = **a**.

Illustrated below with  2 pyramids on top of each other.

So **2h** = **a**, and as the base is square
shaped, Area of Base = **a**** ^{2}**.

This information can be used to form the formula for the volume of a square based pyramid.

Volume of Pyramid = \bf{\frac{a^3}{6}} = \bf{\frac{a \space \times \space a^2}{6}} = \bf{\frac{2h \space \times \space a^2}{6}} = \bf{\frac{2ha^2}{6}} = \bf{\frac{ha^2}{3}},

\bf{\frac{ha^2}{3}} = \bf{\frac{1}{3}} ×

Which is: \bf{\frac{1}{3}} × (Height) × (Area of Base).

Volume =

Fortunately, for a pyramid with a rectangle shaped base, the formula for the volume is the same.

Volume = \bf{\frac{1}{3}} ×

Volume =

Unlike a right pyramid, where the top of the pyramid is above the center of the base.

An oblique pyramid is a pyramid where the top is __NOT__ directly above the center of the base.

The volume formula from before still holds though, as long as it's still the perpendicular height from the ground upwards, that is being used.

Volume = \bf{\frac{1}{3}} ×

Volume =

A pyramid can also have a triangular base, such a shape is called a Tetrahedron.

The approach again is the same as with a pyramid with a square or rectangle base.

Volume = \bf{\frac{1}{3}} × **Area of Base**
× **Height**

For a triangle base pyramid such as the one pictured above, the triangle base looks
like:

The area of this triangle base on its own is given by: \bf{\frac{1}{2}}
× **a** × **b**.

So Volume is:

\bf{\frac{1}{3}} × (\bf{\frac{1}{2}} ×

= \bf{\frac{ab}{6}} ×

This sum can give us the volume of a pyramid with a triangle base.

Volume =

A regular Tetrahedron is a triangular based pyramid where all sides are the same length.

For a regular tetrahedron, the volume can be established by \bf{\frac{\sqrt{2}a^3}{12}}.

Volume =