In mathematics, a pentagonal prism is a three-dimensional geometric figure that has five lateral rectangular faces with two congruent and parallel pentagonal bases.
In this article, we have covered, the definition of Pentagonal Prism, its formula, examples and others in detail.
Table of Content
Pentagonal Prism Definition
A pentagonal prism is a type of heptahedron that belongs to the polyhedron family, which has seven plane faces. It has seven faces, ten vertices, and fifteen edges. The lateral, or side, faces of a pentagonal prism are rectangular-shaped and are connected by two identical pentagonal bases.
Types of Pentagonal Prism
There are three types of pentagonal prisms:
- Regular Pentagonal Prism
- Right Pentagonal Prism
- Oblique Pentagonal Prism
Regular Pentagonal Prism: A pentagonal prism is called a regular pentagonal prism if it has sides that are of the same length.
Right Pentagonal Prism: A right pentagonal prism is a prism that has congruent and parallel pentagonal faces perpendicular to the rectangular faces.
Oblique Pentagonal Prism: An oblique pentagonal prism has pentagonal faces that are not exactly on top of each other, and the rectangular faces are not perpendicular to the pentagonal faces.
Pentagonal Prism Formula
Various pentagonal prism formulas are added below:
Surface Area of a Pentagonal Prism Formula
The surface area of a pentagonal prism is the total area occupied by all its surfaces. The surface area of the prism is equal to the area of its net. So, to determine the surface area of a pentagonal prism, we have to calculate the areas of each of its faces, and then add the resulting areas. A pentagonal prism has two types of surface areas: a lateral surface area and a total surface area.
So, the formula for calculating the lateral surface area (LSA) of a pentagonal prism is given as follows:
Lateral Surface Area of a Pentagonal Prism = 5as square units
where,
- "a" is Apothem Length of Pentagonal Prism
- "s" is Base Length of Pentagonal Prism
Total Surface Area of a Prism (TSA) = LSA + 2 × Base area
So, the formula for calculating the total surface area (TSA) of a pentagonal prism is given as follows:
Total Surface Area of a Pentagonal Prism = (5as + 5sh) square units
where,
- "a" is Apothem Length of Pentagonal Prism
- "s" is Base Length of Pentagonal Prism
- "h" is Height of Pentagonal Prism
Volume of a Pentagonal Prism Formula
The volume of a pentagonal prism is referred to as the space enclosed within a pentagonal prism. The formula for the volume of a pentagonal prism is equal to the product of its base area and its height.
Volume of a Pentagonal Prism (V) = Base Area × Height of the Prism
So, the formula for calculating the volume of a rectangular prism is given as follows:
Volume of a Pentagonal Prism = (5/2) × a × s × h cubic units
where,
- "a" is Apothem Length of Pentagonal Prism
- "s" is Base Length of Pentagonal Prism
- "h" is Height of Pentagonal Prism
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Examples on Pentagonal Prism Formula
Example 1: Find the volume of a pentagonal prism whose apothem length is 5 cm, base length is 9 cm, and height is 12 cm.
Solution:
Given:
- Apothem length of the pentagonal prism (a) = 5 cm
- Base length of the pentagonal prism (s) = 9 cm
- Height of the pentagonal prism, h = 12 cm
We know that,
Volume of a Pentagonal Prism = (5/2) × a × s × h cubic units
= 5/2 × (5 × 9 × 12)
= 5/2 × (540)
= 5 × 270 = 1,350
Therefore, volume of the pentagonal prism is 1650 cm3.
Example 2: Find the height of the pentagonal prism if its volume is 1000 cu. in and its apothem length and base length are 4 in and 8 in, respectively.
Solution:
Given:
- Volume of the pentagonal prism = 1000 cu. in
- Apothem length of the pentagonal prism (a) = 4 in
- Base length of the pentagonal prism (s) = 8 in
We know that,
Volume of a Pentagonal Prism = (5/2) × a × s × h cubic units
⇒ 1000 = (5/2) × 4 × 8 × h
⇒ 1000 = 80h
⇒ h = 1000/80
⇒ h = 12.5 in
Therefore, the height of the pentagonal prism is 12.5 inches.
Example 3: Find the total surface area of the pentagonal prism whose apothem length is 6 in, base length is 10 in, and height is 13 in.
Solution:
Given:
- Apothem length of the pentagonal prism (a) = 6 in
- Base length of the pentagonal prism (s) = 10 in
- Height of the pentagonal prism, h = 13 in
We know that,
Total Surface Area of a Pentagonal Prism = 5as + 5sh square units
= 5 (6 × 10) + 5 (10 × 13)
= 5(60) + 5(150)
= 300 + 750
= 1050 sq. in
Therefore, the total surface area of a pentagonal prism is 1050 sq. inches.
Example 4: Find the lateral surface area of the pentagonal prism whose apothem length is 4 cm, base length is 7 cm, and height is 10 cm.
Solution:
Given:
- Apothem length of the pentagonal prism (a) = 6 in
- Base length of the pentagonal prism (s) = 10 in
- Height of the pentagonal prism, h = 13 in
We know that,
Lateral Surface Area of a Pentagonal Prism = 5as square units
= 5 × 4 × 10
= 200 sq. cm
Therefore, the lateral surface area of a pentagonal prism is 200 sq. cm.
Example 5: Find the volume of a pentagonal prism whose apothem length is 7 cm, base length is 11 cm, and height is 15 cm.
Solution:
Given:
- Apothem length of the pentagonal prism (a) = 7 cm
- Base length of the pentagonal prism (s) = 11 cm
- Height of the pentagonal prism, h = 15 cm
We know that,
Volume of a Pentagonal Prism = (5/2) × a × s × h cubic units
= 5/2 × (7 × 11 × 15)
= 2,887.5 cm3
Therefore, the volume of the pentagonal prism is 2,887.5 cm3.
Practice Questions on Pentagonal Prsim Formula
Q1. Find the surface area of a pentagonal prism with a base edge length of 4 cm, an apothem length of 2.8 cm, and a height of 10 cm.
Q2. Calculate the volume of a pentagonal prism with a base edge length of 5 cm, an apothem length of 3.4 cm, and a height of 12 cm.
Q3. Determine the lateral surface area of a pentagonal prism with a base edge length of 6 cm and a height of 15 cm.
Q4. Calculate the volume of a pentagonal prism with a base edge length of 7 cm, an apothem length of 5 cm, and a height of 20 cm.