# Volume of a Square Pyramid

The volume of a square pyramid is the space it occupies in a 3-dimensional plane. The volume is the capacity of a square pyramid or the number of unit cubes that can be fit into it. It is expressed in cubic units such as m3, cm3, mm3, and in3.

## Formula

The formula to calculate the volume of a right square pyramid is the same as that of a non-right square pyramid as we consider the perpendicular height of the pyramid for both cases.

The formula is:

Let us solve some examples to understand the above concept better.

## Solved Examples

Find the volume of a square pyramid with a base of 16 cm, and a height of 21 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 16 cm, h = 21 cm
∴ V = ${\dfrac{1}{3}\times 16^{2}\times 21}$
= 1792 cm3

Find the volume of a square based pyramid with a base of 4 in, and a height of 6 in.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 4 in, h = 6 in
∴ V = ${\dfrac{1}{3}\times 4^{2}\times 6}$
= 32 in3

Find the volume of a right square pyramid with a base of 7 cm, and a height of 15 cm.

Solution:

As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 7 cm, h = 15 cm
∴ V = ${\dfrac{1}{3}\times 7^{2}\times 15}$,
= 245 cm3

We might have to calculate the volume of a square pyramid when the height is not given directly, but instead the slant height and base may be known. We generally calculate the height using the Pythagorean Theorem and then calculate the volume.

Let us solve an example to understand the concept better.

Finding the volume of a square pyramid when the BASE and SLANT HEIGHT are known

Find the volume of a pyramid with a square base given in the figure with its base and slant height.

Solution:

Here we will use the slant height to find the perpendicular height and then calculate the volume.
${s^{2}=h^{2}+\left( \dfrac{b}{2}\right)^{2}}$, applying Pythagorean Theorem
s2 = h2 + (b/2)2
∴ ${h=\sqrt{s^{2}-\left( \dfrac{b}{2}\right) ^{2}}}$, here b = 6 cm, s = 9 cm
= ${\sqrt{9^{2}-3^{2}}}$
= 8.48 cm
As we know,
Volume (V) = ${\dfrac{1}{3}b^{2}h}$, here b = 6 cm, h = 8.48 cm
∴ V = ${\dfrac{1}{3}\times 6^{2}\times 8.48}$
= 101.76 cm3

Finding the volume of a truncated square pyramid when the BASE SIDE, TOP SIDE, and the HEIGHT are known

Find the volume of a given truncated square pyramid.

Solution:

Here, we will use the volume formula for a truncated square pyramid.
Volume (V) = ${\dfrac{1}{3}\times h\times \left( a^{2}+b^{2}+ab\right)}$, here a = 5 cm, b = 3 cm, h = 4 cm
∴ V = ${\dfrac{1}{3}\times 4\times \left( 5^{2}+3^{2}+5\times 3\right)}$
= 65.33 cm3