# Volume of a Pentagonal Prism

The volume of a pentagonal prism is the space it occupies in the three-dimensional plane. It is measured in cubic units such as m3, cm3, mm3, ft3.

## Formula

The formula is given below:

Let us solve some examples to understand the concept better.

## Solved Examples

Find the volume of a regular pentagonal prism given in the figure.

Solution:

As we know,
Volume (V) = ${\dfrac{5}{2}abh}$,, here, a = 6.193 m, b = 9 m, h = 20 m
${\therefore V= \dfrac{5}{2}\times 6.193\times 9\times 20}$
= 2786.85 m3

Finding the volume of a pentagonal prism when the BASE EDGE and HEIGHT are known

Find the volume of a regular pentagonal prism with a base edge of 4 cm and a height of 7 cm.

Solution:

Here we will use an alternative formula.
Volume (V) = ${\dfrac{1}{4}\sqrt{5\left( 5+2\sqrt{5}\right) }b^{2}h}$ , here b = 4 cm, h = 7 cm
${\therefore V= \dfrac{1}{4}\sqrt{5\left( 5+2\sqrt{5}\right) }\times 4^{2}\times 7 }$
≈ 192.7 cm3

Finding the volume of an oblique pentagonal prism when the BASE EDGE and HEIGHT are known

Find the volume of the oblique pentagonal prism given in the figure.

Solution:

As we know,
The volume of an oblique pentagonal prism = the volume of a right pentagonal prism having the height ‘h’
Using the alternative formula
Volume (V) = ${ \dfrac{1}{4}\sqrt{5\left( 5+2\sqrt{5}\right) }b^{2}h }$ , here b = 6 m, h = 13 m
${\therefore V= \dfrac{1}{4}\sqrt{5\left( 5+2\sqrt{5}\right) }\times 6^{2}\times 13 }$
= 805.2 m3

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