Volume of a Pentagonal Prism – Definition, Formula, Examples, Facts

What Is the Volume of a Pentagonal Prism?

The volume of a pentagonal prism is the amount of space occupied by it. It is calculated by the formula: Base $\times$ area Height. Volume is measured in cubic units such as $inch^{3}$, $ft^{3}$, etc.

A pentagonal prism is a three-dimensional solid with two bases (top and bottom) that are pentagons. A pentagonal prism has sides (lateral faces) that are rectangular. Volume of a pentagonal prism is defined as the space occupied within the boundaries of the prism in three-dimensional space.

Volume of a Pentagonal Prism Formula

The volume of a prism is calculated by multiplying the area of the base by the height of the prism.

Volume of a pentagonal prism $=$ Base area $\times$ Height
$V = \frac{5}{2} \times a \times b \times h$ where, $a =$ length of apothem of the pentagonal base $b =$ base edge (length of sides) of the pentagonal prism $h =$ the height of the pentagonal prism

Here, we use the following formulas:

• Base Area of Pentagonal Prism $=$ Area of a Pentagon 

Base Area $= (12 \times \text{Perimeter}) \times \text{Apothem}$

• Perimeter of a Regular Pentagon $= P = 5 \times b$                   

where b is the side length

Keep in mind:

• The apothem is the line segment joining the midpoint of one of the sides and the center of the polygon. 
• Perimeter of a pentagon is equal to the sum of all its sides.

How to Find the Volume of a Pentagonal Prism

When the base-edge (b), height of the prism (h), and apothem of the base (a) is known, we use the following steps to calculate the volume.

Step 1: Find the base area of the given pentagonal prism. 

Base area $=$ Area of pentagon 

Base area $= (\frac{1}{2} \times \text{Perimeter}) \times \text{Apothem}$

Base area $= (\frac{1}{2} \times 5b) \times a$                  Perimeter of a regular pentagon $= 5b$

Step 2: Calculate the required volume using the formula: 

Volume of a pentagonal prism $= \frac{1}{2} \times a \times 5b \times h$ cubic units.

Volume of a pentagonal prism $= \frac{5}{2}\;abh$

where

$a =$  apothem length of the pentagonal prism.

$b =$  the base length of the pentagonal prism.

$h =$  the pentagonal prism’s height.

Example: Find the volume of a pentagonal prism with base-edge $= 7$ units, apothem $(a) ≈ 4.8$ units, height of the prism (h) $= 10$ units.

$a = 4.8$ units

$b = 7$ units

$h = 10$ units

$V = \frac{5}{2}abh = \frac{5}{2} \times 7 \times 4.8 \times 10 = 840$ cubic units 

Facts about Volume of a Pentagonal Prism

Conclusion

In this article, we have discussed the volume of a pentagonal prism and the formula for the same. Now let’s look at some solved examples and do some practice problems to understand the concept better.

Solved Examples on Volume of a Pentagonal Prism

1. Find the volume of a pentagonal prism whose apothem length of 3 inches, base length of 12 inches, and height of 15 inches.

Solution: 

Apothem length (a) $= 3$ inches

Base length $(b) = 12$ inches

Height $(h) = 15$ inches

Volume of the pentagonal prism $= \frac{5}{2} \times a \times b \times h$

    $= \frac{5}{2} \times 3 \times 12 \times 15$

   $= 1350\; inch^{3}$

Hence, the volume of a pentagonal prism $= 1350\; inch^{3}$

2. Find the apothem length of the pentagonal prism if the height is 20 feet, the base length is 7 feet, and its volume is $1680\; ft^{3}$.

Solution:

Volume of pentagonal prism $= 1240\; ft^{3}$

$h = 20$ feet

$b = 7$ feet

Volume of pentagonal prism $= \frac{5}{2} \times a \times b \times h$

$\Rightarrow 1680 = \frac{5}{2} \times a \times 7 \times 20$

$\Rightarrow a = \frac{2 \times 1680}{5 \times 7 \times 20} = 4.8$ feet

Hence, apothem length of the pentagonal prism $= 4.8$ feet

3. If the volume of a pentagonal prism is $528\; ft^{3}$ and the base area is $24\; ft^{2}$, then find the height of the prism.

Solution:

Volume $= 528$ cubic feet and base area $= 24$ square feet

Volume of prism $= base area \times height$

  $\Rightarrow 528 =24 \times height$

$ \Rightarrow height = \frac{528}{24} = 22\; ft$

Hence, the height of the prism $= 22\; ft$

4. Find the base area of a pentagonal prism if apothem length of 3.4 units and its perimeter is 25 units.

Solution:

Apothem length $(a) = 3.4$ units, and perimeter $(P) = 28$ units

Base area of a pentagonal prism $= \frac{1}{2} \times P \times a$

$= \frac{1}{2} \times 25 \times 3.4$

$= 42.5$ square units

Hence, the base area of a pentagonal prism $= 42.5$ square units

5. Find the height of the pentagonal prism if the apothem length is 4 inches, the base length is 6 inches and its volume is 540 cubic inches.

Solution:

apothem length $(a) = 4$ inches

base-edge $= 6$ inches 

Volume $= 540$ cubic inches

Volume of the pentagonal prism $= \frac{5}{2} \times a \times b \times h$

$\Rightarrow 540 = \frac{5}{2} \times 4 \times 6 \times h$

$\Rightarrow 540 = 60\; h$

$\Rightarrow h = \frac{540}{60}$

$\Rightarrow h = 9$ inches

Hence, the height of a pentagonal prism $= 9$ inches

Practice Problems on Volume of a Pentagonal Prism

Frequently Asked Questions on Volume of a Pentagonal Prism