# Pentagonal Prism

## What is a Pentagonal Prism?

• A prism that consists of two pentagonal bases the top and the bottom and has five rectangular sides is known as a pentagonal prism.

• A pentagonal prism is a type of heptahedron with 15 edges, 10 vertices, and 7 faces.

• A pentagonal prism has pentagonal bases which give five sides of the prism

• The other name for a pentagonal prism is a five-sided polygon prism.

### What is a Prism?

In mathematics, a prism is generally a three-dimensional box that is a solid figure with a uniform cross-section and it has two common bases.

Face:  Faces of the pentagonal prism can be defined as the flat side of a 3-dimensional object.

Base:  The base is defined as one of two congruent and parallel sides of an object.

Edge:  An edge is a line of intersection of two faces on a solid object.

Vertex:  Vertex is the joining point where two edges of a prism meet.

### Properties of a Pentagonal Prism:

Pentagonal prism faces, edges, vertices have been listed below in a tabular form.

 Number of faces (Prism Faces) 7 Number of Edges 15 Number of Vertices 10 Base shape Pentagon Side shape Rectangle Faces by sides 5+2

Here’s a diagram showing Pentagonal prism faces, edges, vertices-

### Types of Pentagonal Prisms-

The two types of Pentagonal Prisms are –

1. Regular Pentagonal Prisms

2. Right Pentagonal Prisms

### What is a Regular Pentagonal Prism?

• In a regular pentagonal prism, all the sides of the pentagonal prism are of exactly equal length.

• All the rectangular faces of pentagonal prisms are congruent.

• The rectangular faces in a regular pentagonal prism are said to be lateral when the pentagonal faces of the regular pentagonal prism are bases.

• The lateral side faces of pentagonal prisms are known as lateral edges.

### What is a Rectangular Pentagonal Prism?

• If a prism has two congruent and parallel pentagonal faces and five rectangular faces which are perpendicular to the triangular ones of the pentagonal prism are known as Rectangular Pentagonal prism.

### Surface Area of a Pentagonal Prism-

The surface area of a prism can be defined as the sum of the areas of the lateral faces of a prism and its bases.

Length(a) = Apothem of the pentagonal prism

Length(b) = Base of the pentagonal prism

Length(h) = Height of the pentagonal prism

Let’s find the surface area of the pentagonal prism using the given information,

 Surface area of the Pentagonal Prism = 5×a ×b +5×b×h square       units

### Volume of a Pentagonal Prism-

The volume of a pentagonal prism can be defined as the amount of space inside the pentagonal prism.

In the diagram given above,

Length(a) = Apothem of the pentagonal prism

Length(b) = Base of the pentagonal prism

Length(h) = Height of the pentagonal prism

Let’s find the volume of the pentagonal prism,

 Volume of the Pentagonal Prism = (5/2) ×a ×b ×h cubic units

### Questions on Pentagonal prism-

Question 1) What are prism faces and define pentagonal prism faces?

Solution) Faces of a pentagonal prism can be defined as a flat side of a 3-dimensional object. There are seven prism faces.

Question 2) Calculate the surface area and the volume of the pentagonal prism with the given measurements as apothem length equal to 4 cm, base length equal to 8 cm, and height equal to 10 cm?

Solution) Let’s list down the given information,

Apothem Length(a) of the pentagonal prism, a = 4 cm

Base length (b) of the pentagonal prism, b = 8 cm

Height (h)of the pentagonal prism, h = 10 cm

We know the formula for the volume of a pentagonal prism,

Volume of the Pentagonal Prism = (5/2) ×a ×b ×h cubic units

= 5/2 × (4×8×10)

= 5/2 × (320)

= 5 × (160)

= 800 cubic units

Therefore, the volume of the given pentagonal prism is 800 cm3.

We know the formula for the surface area of a pentagonal prism,

Surface area of the Pentagonal Prism = 5×a ×b +5×b×h square units

= 5 (4×8) + 5 (8×10)

= 5(32) + 5(80)

= 160 + 400

= 560

Therefore, the surface area of the given pentagonal prism is 560 cm2.

Question 3) Calculate the volume of a pentagonal prism with the given measurements as apothem length equal to 10 cm, base length equal to 12 cm, and height equal to 16 cm?

Solution) Let’s list down the given information,

Apothem Length(a) of the pentagonal prism, a = 10cm

Base length (b) of the pentagonal prism, b = 12 cm

Height (h)of the pentagonal prism, h = 16 cm

We know the formula for the volume of a pentagonal prism,

Volume of the Pentagonal Prism = (5/2) ×a ×b ×h cubic units

= 5/2 × (10×12×16)

= 5/2 × (1920)

= 5 × (960)

= 4,800 cubic units

Therefore, the volume of the given pentagonal prism is 4,800 cm.

FAQ (Frequently Asked Questions)

1. What is a Pentagonal Based Prism?

Here’s what a Pentagonal Prism is-

A prism that consists of two pentagonal bases the top and the bottom and has five rectangular sides is known as a pentagonal prism.

A pentagonal prism is a type of heptahedron with 15 edges, 10 vertices and 7 faces.

A pentagonal prism has pentagonal bases which give five sides of the prism

The other name for a pentagonal prism is a five-sided polygon prism.

2. What do you mean by a Prism?

A prism can be defined as a three-dimensional shape which generally consists of two parallel and congruent faces known as bases.

3. Give the Formula for the Surface Area of a Pentagonal Prism?

We can find the surface area of a pentagonal using the following formula:

Surface area of the Pentagonal Prism = 5×a ×b +5×b×h square units

4. What is the Formula for the volume of a Pentagonal Prism?

We can find the volume of a pentagonal using the following formula:

Volume of the Pentagonal Prism = (5/2) ×a ×b ×h cubic units

5. Is Cube a Prism?

A cube is a special case of square prism and a square prism is a special case of a rectangular prism. All of these are cuboids.

How many Edges does a Prism have?

Here’s the answer to how many edges does a prism have,

 Type of Prism Edges Triangular prism 9 Rectangular prism 12

The question of how many edges does a prism have is at least 9, but the number of edges varies with the type of prism.