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Pentagonal Prism
In geometry, a pentagonal prism is a threedimensional shape with two pentagonal bases and five rectangular faces. So, a pentagonal prism has a total of 7 faces, 15 edges, and 10 vertices out of which 2 faces are pentagonal in shape. We can find the volume and surface area of a pentagonal prism as we do for any other 3D shape. Let's learn everything about pentagonal prism in detail in this article.
1.  What is a Pentagonal Prism? 
2.  Properties of a Pentagonal Prisms 
3.  Types of Pentagonal Prisms 
4.  Pentagonal Prism Formulas 
5.  Net of a Pentagonal Prism 
6.  FAQs on Pentagonal Prism 
What is a Pentagonal Prism?
A pentagonal prism is a threedimensional solid that has two pentagonal bases  bottom and top. All the other sides of a pentagonal prism have the shape of a rectangle. It is easy to understand the shape of a pentagonal prism by drawing a pentagon on a piece of paper using straight lines. Then imagine it stretching up from the sheet of paper. The 3D shape so formed will be a pentagonal prism!
A pentagonal prism has:
 7 faces
 15 edges
 10 vertices
 Pentagon bottom and top
Pentagonal Prism Examples
Here are some reallife examples of a pentagonal prism. Look at the figures given below and look around you to find more examples of pentagonal prism and make your own list.
 U.S. Defence Department Headquarter
 Barns and boxes
 Decorative items
Properties of a Pentagonal Prism
A pentagonal prism consists of various properties that are unique to this solid shape. Listed below are some of the properties of a pentagonal prism:
 A pentagonal prism has 15 edges, 7 faces, and 10 vertices.
 The base of a pentagonal prism is in the shape of a pentagon.
 The sides of a pentagonal prism are shaped like a rectangle.
 A pentagonal prism is a type of heptahedron, which is a polyhedron with seven plane faces.
 A pentagonal prism can also be called a fivesided polygon prism.
 The crosssection of a pentagonal prism is a pentagon.
Types of Pentagonal Prisms
Pentagonal prisms can be categorized into the following types:
 Regular pentagonal prisms
 Right pentagonal prisms
 Oblique pentagonal prisms
Regular Pentagonal Prisms
If all the sides of a pentagonal prism are equal in length, it is called a regular pentagonal prism. Note the following properties of a regular pentagonal prism.
 All rectangular faces of a regular pentagonal prism are congruent.
 When the pentagonal faces of a regular pentagonal prism are the bases, the rectangular faces of the prism are said to be lateral.
Right Pentagonal Prism
If the pentagon faces of a pentagonal prism are congruent and parallel, and the rectangular faces are perpendicular to the pentagon faces, it is called a right pentagonal prism.
Oblique Pentagonal Prism
If the pentagon faces of a pentagonal prism are not exactly on top of each other, that is, when the rectangular faces are not perpendicular to the pentagon faces, it's called an oblique pentagonal prism.
Pentagonal Prism Formulas
The formulas of a pentagonal prism help us to find out the surface area and its volume very quickly and easily. Observe the following figure where 'a' is the apothem length of the prism, 'b' is the base length, and 'h' is the height of the prism.
Surface Area of a Pentagonal Prism
The surface area of a pentagonal prism gives the area of each face of the prism. The formula for the surface area of a pentagonal prism is:
Surface Area = (5ab + 5bh) square units
Pentagonal Prism Volume
The volume of a pentagonal prism gives the capacity of the prism. The formula for the volume of a pentagonal prism is:
Volume = (5/2 × abh) cubic units
Example: If the apothem length 'a' of a pentagonal prism is 5 feet, the base length 'b' is 4 feet, and the height 'h' is 6 feet.
The surface area of the pentagonal prism is: 5ab + 5bh = (5 × 5 × 4) + (5 × 4 × 6) = 100 + 120 = 220 square feet.
The volume of the pentagonal prism is: 5/2 × abh = 5/2 × 5 × 4 × 6 = 300 cubic feet.
Net of a Pentagonal Prism
The net of a pentagonal prism is the flattened view of the shape. When the pentagonal prism is opened, the two shapes that can be seen are the rectangle and the pentagon. The net of the pentagonal prism consists of the two pentagons which is the base of the shape and rectangles which is the sides or faces of the shape.
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Pentagonal Prism Examples

Example 1: Jenifer is packing a chocolate box in the shape of a pentagonal prism. She needs wrapping paper to pack it. Determine the surface area of the wrapping paper needed if the dimensions of the box are as follows:
 Apothem length, a = 10 inches
 Base length, b = 20 inches
 Height, h = 15 inches
Solution:
To determine the surface area of the wrapping paper, we need to determine the surface area of the box. The surface area of the box is 5ab + 5bh = (5 × 10 × 20) + (5 × 20 × 15) = 1000 + 1500 = 2500 square inches.
Therefore, the surface area of the wrapping paper 2500 square inches.

Example 2: Emily wants to fill a pentagonal prismshaped container with water. Determine the capacity of the container so that Emily knows how much water is needed. The dimensions of the container are as follows:
 Apothem length, a = 25 feet
 Base length, b = 17 feet
 Height, h = 15 feet
Solution:
To know the capacity of the container, we need to find the volume of the container. The volume of the pentagonal prismshaped container is 5/2 × abh = 5/2 × 25 × 17 × 15 =15,937.5 feet^{3}.
Therefore, the capacity of the container is 15,937.5 feet^{3}.

Example 3: Find the height of the pentagonal prism if the total surface area of the pentagonal prism is 90 sq. yd., apothem length and base length are 4 yds. and 2 yds. respectively.
Solution: Given: Total surface of pentagonal prism, TSA = 90 sq yd, a = 4 yd and b = 2 yd
We know that the total surface area of a pentagonal prism = 5ab + 5bh = 5b(a+h) = 90
⇒ 10(4+h) = 90 ⇒ 4 + h = 90/10Thus, h = 9  4
⇒ h = 5 ydsTherefore, the height of the pentagonal prism is 5 yds.
FAQs on Pentagonal Prism
What is a Pentagonal Prism?
A pentagonal prism is a threedimensional solid that has two pentagonal bases on the bottom and top of the shape. All the other sides of a pentagonal prism have the shape of a rectangle. There are types of pentagonal prism  regular pentagonal prism and rectangular pentagonal prism. A pentagonal prism has 15 edges, 7 faces, and 10 vertices.
What are the Properties of a Pentagonal Prism?
Listed below are a few of the properties of a pentagonal prism:
 The base of a pentagonal prism is in the shape of a pentagon.
 The sides of a pentagonal prism are shaped like a rectangle.
 A pentagonal prism is a type of heptahedron, which is a polyhedron with seven plane faces.
 A pentagonal prism can also be called a fivesided polygon prism.
 The crosssection of a pentagonal prism is a pentagon.
What does an Octagonal Prism look like?
An octagonal prism is a threedimensional solid. It has two opposite and parallel octagon bases and eight rectangular faces.
What is the Definition of a Prism?
A prism is a 3dimensional shape with two identical shapes facing each other. These identical shapes are called bases, which can be a square, triangle, or any polygon. The other faces of a prism are usually parallelograms and rectangles.
What is a 3D Pentagon called?
A threedimensional shape that includes pentagons is a pentagonal prism. It includes 2 pentagonal bases and 5 rectangular faces.
What is the Formula for the Surface Area of a Pentagonal Prism?
The surface area of a pentagonal prism can be found out using the following formula: A = 5ab + 5bh.
What is the Formula for the Volume of a Pentagonal Prism?
The volume of a pentagonal prism can be calculated using the following formula: V = 5/2 × a × b × h.
How many Vertices does a Pentagonal Prism have?
There are ten vertices in a pentagonal prism, at both the pentagonal bases of the figure. 5 vertices are there on the top face and five vertices are there on the bottom face.
Is a Cylinder a Prism?
A prism is a polyhedron that has only flat faces. But a cylinder has curved surfaces, therefore, it is not a prism.
What is the Net of a Pentagonal Prism?
The net of a pentagonal prism consists of 2 pentagonshaped faces located at the top and bottom of the shape along with 5 rectangleshaped sides. The top and bottom pentagons are the bases and the rectangles are the faces.
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