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In mathematics, a pentagonal prism is a three-dimensional geometric figure that has five lateral rectangular faces with two congruent and parallel pentagonal bases. 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. There are three types of pentagonal prisms: regular pentagonal prism, right pentagonal prism, and oblique pentagonal prism. A pentagonal prism is called a regular pentagonal prism if it has sides that are of the same length. A right pentagonal prism is a prism that has congruent and parallel pentagonal faces perpendicular to the rectangular faces. 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

## 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 the apothem length of the pentagonal prism, and
s” is the base length of the pentagonal prism.

The 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 the apothem length of the pentagonal prism
s” is the base length of the pentagonal prism
h” is the height of the 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 the apothem length of the pentagonal prism
s” is the base length of the pentagonal prism
h” is the height of the prism.

## Solved 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 data:

Apothem length of the pentagonal prism (a) = 5 cm

The base length of the pentagonal prism (s) = 9 cm

Height of the pentagonal prism, h = 12 cm

We know that,

The 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, the 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:

The volume of the pentagonal prism = 1000 cu. in

Apothem length of the pentagonal prism (a) = 4 in

The base length of the pentagonal prism (s) = 8 in

We know that,

The 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 data,

Apothem length of the pentagonal prism (a) = 6 in

The base length of the pentagonal prism (s) = 10 in

Height of the pentagonal prism, h = 13 in

We know that,

The 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 data,

Apothem length of the pentagonal prism (a) = 6 in

The base length of the pentagonal prism (s) = 10 in

Height of the pentagonal prism, h = 13 in

We know that,

The 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 data:

Apothem length of the pentagonal prism (a) = 7 cm

The base length of the pentagonal prism (s) = 11 cm

Height of the pentagonal prism, h = 15 cm

We know that,

The 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.

## FAQs on Pentagonal Prism Formula

Question 1: What is a pentagonal prism?

In mathematics, a pentagonal prism is a three-dimensional geometric figure that has five lateral rectangular faces with two congruent and parallel pentagonal bases. It has seven faces, ten vertices, and fifteen edges.

Question 2: What is the formula for calculating the total surface area of a pentagonal prism?

The formula for calculating the total surface area (TSA) of a pentagonal prism is given as follows:

TSA = (5as + 5sh) square units

where,
“a” is the apothem length of the pentagonal prism
“s” is the base length of the pentagonal prism
“h” is the height of the prism.

Question 3: What is the formula for calculating the volume of a pentagonal prism?

The volume of a pentagonal prism is referred to as the space enclosed within a pentagonal prism. 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 the apothem length of the pentagonal prism
“s” is the base length of the pentagonal prism
“h” is the height of the prism.

Question 4: What is the formula for calculating the lateral surface area of a pentagonal prism?

The formula for calculating the lateral surface area (LSA) of a pentagonal prism is given as follows:

LSA = 5as square units

where,
“a” is the apothem length of the pentagonal prism
“s” is the base length of the pentagonal prism

Question 5: What are the different types of a pentagonal prism?