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.
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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
- 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
Volume of a Pentagonal Prism - Definition, Formula, Examples, Facts
Volume of a pentagonal prism is given by
Volume of a pentagonal prism $= \text{base area} \times \text{height}$
The volume of a pentagonal prism with base-edge b, apothem a, and height h is _________.
Volume of the pentagonal prism $= \frac{5}{2} \times a \times b \times h$
The volume of a pentagonal prism whose base area is $50\; ft^{2}$ and height is 5 ft.
Volume of pentagonal prism $=base area \times height = 50 \times 5 = 250\; ft^{3}$
The base area of a pentagonal prism of apothem length is 2 feet and its perimeter is 15 feet is _____.
Base area of a pentagonal prism $= \frac{1}{2} \times perimeter \times apothem length$
$\Rightarrow$ Base area of a pentagonal prism $= \frac{1}{2} \times 15 \times 2 = 15\; ft^{2}$
The volume of a pentagonal prism is ______.
Volume of the pentagonal prism is the total space occupied by it.
Frequently Asked Questions on Volume of a Pentagonal Prism
What is a pentagonal prism?
A pentagonal prism is a three-dimensional solid with five rectangular lateral faces and two pentagonal bases (top and bottom).
What is the formula for the surface area of a pentagonal prism?
Surface area of pentagonal prism $= 5ab + 5bh$ square units
What is an oblique pentagonal prism?
An oblique prism is one whose sides don’t meet the base at a right angle.
What is the apothem length of a polygon?
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.
