# 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.

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

• A prism with five rectangular lateral faces (sides) and two pentagonal bases (top and bottom) is called a pentagonal prism.
• A pentagonal prism is a kind of heptahedron that has 15 edges, 10 vertices, and 7 faces.
• A pentagonal prism has a pentagonal cross-section.
• In the right pentagonal prism, the bases are aligned exactly on top of each other.
• When the base-edge (b) and and the height of a right regular pentagonal prism (h) is known, we can use the volume formula $V = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})}b^{2} h ≈ 1.72 b^{2} h$
Example: Find the volume of a pentagonal prism with base-edge $= 7$ units and height $= 10$ units. $V = \frac{1}{4}\sqrt{5(5 + 2\sqrt{5})} b^{2} h ≈ 1.72 b^{2} h ≈ 842.8$ cubic units

## 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

1

### Volume of a pentagonal prism is given by

$\text{apothem} \times \text{height}$
$\text{base perimeter} \times \text{height}$
$\text{base area} \times \text{height}$
$\text{Total surface area} \times \text{height}$
CorrectIncorrect
Correct answer is: $\text{base area} \times \text{height}$
Volume of a pentagonal prism $= \text{base area} \times \text{height}$
2

### The volume of a pentagonal prism with base-edge b, apothem a, and height h is _________.

$\frac{1}{2} \times a \times b \times h$
$\frac{5}{2} \times a \times b \times h$
$\frac{2}{5} \times a \times b \times h$
$\frac{5}{2} \times a \times h$
CorrectIncorrect
Correct answer is: $\frac{5}{2} \times a \times b \times h$
Volume of the pentagonal prism $= \frac{5}{2} \times a \times b \times h$
3

### The volume of a pentagonal prism whose base area is $50\; ft^{2}$ and height is 5 ft.

$250\; ft^{3}$
$299\; ft^{3}$
$235\; ft^{3}$
$280\; ft^{3}$
CorrectIncorrect
Correct answer is: $250\; ft^{3}$
Volume of pentagonal prism $=base area \times height = 50 \times 5 = 250\; ft^{3}$
4

### The base area of a pentagonal prism of apothem length is 2 feet and its perimeter is 15 feet is _____.

$38\; ft^{2}$
$30\; ft^{2}$
$55\; ft^{2}$
$15\; ft^{2}$
CorrectIncorrect
Correct answer is: $15\; ft^{2}$
Base area of a pentagonal prism $= \frac{1}{2} \times perimeter \times apothem length$
$\Rightarrow$ Base area of a pentagonal prism $= \frac{1}{2} \times 15 \times 2 = 15\; ft^{2}$
5

### The volume of a pentagonal prism is ______.

The region covered by all its surfaces
Total area of the lateral faces
Total space occupied by it
Total area of the base
CorrectIncorrect
Correct answer is: Total area of the lateral faces
Volume of the pentagonal prism is the total space occupied by it.

## Frequently Asked Questions on Volume of a Pentagonal Prism

A pentagonal prism is a three-dimensional solid with five rectangular lateral faces and two pentagonal bases (top and bottom).

Surface area of pentagonal prism $= 5ab + 5bh$ square units

An oblique prism is one whose sides don’t meet the base at a right angle.

Apothem is the line segment that joins any side’s midpoint and the center of a polygon is called the apothem length. It makes a right angle at the point of intersection with the side.