How To Find The Volume Of A Pentagon

Volume of Pentagonal Prism

The volume of a pentagonal prism is the space occupied by it. A prism is a three-dimensional shape having identical bases, flat rectangular side faces, and the same cross-section all along its length. Prisms are classified into different types and named as per the shape of their base. A pentagonal prism as the name says is a three-dimensional solid that has two pentagonal bases: bottom and top. The sides of a pentagonal prism are rectangular in shape. There is a formula to find the volume of a pentagonal prism. Let's explore and learn in detail!

1.	What is Volume of Pentagonal Prism?
2.	Volume of Pentagonal Prism Formula
3.	How To Calculate the Volume of Pentagonal Prism?
4.	FAQs on Volume of Pentagonal Prism

What is Volume of Pentagonal Prism?

The volume of the pentagonal prism is defined as the capacity of the pentagonal prism. In geometry, a pentagonal prism is a three-dimensional shape with two pentagonal bases and five rectangular faces. So, a pentagonal prism has a total of 7 faces(out of which 2 faces are pentagonal in shape), 15 edges, and 10 vertices. By definition, there are 2 pentagonal bases that are congruent to each other. It has 5 lateral faces of rectangle shape, whereas its bottom and top are of pentagon shape. The volume of the pentagonal prism is measured in cubic units such as cm³, m³, in³, etc.

Volume of Pentagonal Prism Formula

We will see the formulas to calculate the volumes of different types of pentagonal prisms. The volume of any prism is obtained by multiplying its base area by its height. i.e., the volume of a prism = base area × height. We will use this formula to calculate the volume of a pentagonal prism as well. We know that the base of a pentagonal prism is a pentagon. By applying the above formula, the volume of a pentagonal prism = area of base × height.

The volume of a pentagonal prism = area of base × height

The volume of a pentagonal prism determines the capacity of the prism. As per the general formula of the volume of a prism, that is, volume = area of base × height. The area of base = 1/2 × Perimeter × Apothem sq units, where perimeter = 5b. Thus, the formula for the volume of a pentagonal prism is: Volume = (5/2 × abh) cubic units where,

a = Apothem length of the pentagonal prism
b = Base length of the pentagonal prism
h = Height of the pentagonal prism

Volume of Pentagonal Prism

Since the third dimension of measurements has come into the picture, thus the unit will be cubic units like these cubic centimeters. There are three types of pentagonal prisms - regular pentagonal prisms, right pentagonal prisms, and oblique pentagonal prisms.

Volume of Pentagonal Prism as per types of pentagonal prisms

In the case of a regular pentagonal prism, both the pentagonal sides are of equal length and the five rectangular sides with bases being perpendicular to each other.
In the case of a right pentagonal prism, the bases are perpendicular to each other.
In the case of an oblique pentagonal prism, the bases are not perpendicular to each other. Thus, the perpendicular drawn from one vertex of one base to the other base of the prism will be taken as its height.

How To Calculate the Volume of Pentagonal Prism?

Here are the steps to calculate the volume of a pentagonal prism. We need to be sure that all measurements are of the same units. Refer to the example given below followed by the steps.

Step 1: Identify the apothem length and base length and find its area using a suitable formula(base area= 5/2ab).
Step 2: Identify the given height of the prism (It should be the height of the total prism).
Step 3: Find the product of its base area and the height to find the volume.

Example: Calculate the volume of the pentagonal prism 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.

Solution: Given that a = 5 feet, b = 4 feet and h = 6 feet. The volume of the pentagonal prism is obtained using the formula V = 5/2 × abh. The steps to determine the volume of the pentagonal prism are:

Step 1: The area of the base of the pentagonal prism is found using the formula, 5/2ab = 5/2 × 5 × 4 = 50 square feet.
Step 2: The height of the prism is 6 ft.
Step 3: The volume of the pentagonal prism = base area × height = 50 × 6 = 300 cubic feet.

Thus, the volume of the pentagonal prism is 300 cubic feet.

Examples on Volume of Pentagonal Prism

Example 1: Find the height of the pentagonal prism if the apothem length is given as 16 units, the base length is 9 units and the volume is 2520 cubic units.

Solution: Given, apothem length, a = 16 units, base length, b = 9 units, volume, v = 2520 units.

We know that the volume of a pentagonal prism = 5/2abh = 5/2 × 16 × 9 × h = 2520
⇒ 360h = 2520.

Therefore, the height of the pentagonal prism, h = 2520/360 = 7 units.
Example 2: Ron has to determine the capacity of a container that is pentagonal prism-shaped. So he decided to fill it with water up to the brim. Calculate the volume of the container which has, apothem length, a = 20 feet, base length, b = 12 feet, and height, h = 10 feet

Solution: To know the capacity of the container, Ron needs to find the volume of the container. Given that a = 20 feet, b = 12 feet and h = 10 feet. The volume of the pentagonal prism-shaped container is:

V = 5/2 × abh
⇒ V = 5/2 × 20 × 12 × 10 = 6000 cubic feet.

Therefore, the capacity of the container is 6000 cubic feet.

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Practice Questions on Volume of Pentagonal Prism

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FAQs on Volume of Pentagonal Prism

What is the Volume of a Pentagonal Prism?

The volume of a pentagonal prism is the amount of space it can occupy, which can be found out using the pentagonal prism volume formula, which is the product of its base area by its height. The volume of a pentagonal prism can be expressed in cubic units such as m³, in³, etc.

What is the Formula for Calculating the Volume of a Pentagonal Prism?

We calculate the volume of a pentagonal prism using the formula is V = 5/2abh where this formula is further understood as V = [1/2 × 5 × base length × apothem] × height of the prism. The values are put in the formula and write the answer so obtained in cubic units.

How to Find the Volume of a Regular Pentagonal Prism?

To find the volume of a regular pentagonal prism,

Step 1: Identify the values of the apothem length, base length, and height of the respective pentagonal prism.
Step 2: Put the values in the formula, v = 5/2abh, and obtain the value of the volume of the regular pentagonal prism.

How to Calculate the Volume of a Pentagonal Prism?

To calculate the volume of a pentagonal prism, we need to follow the steps given below:

Step 1: Check for the given information like the base, apothem, and height.
Step 2: Put the given values in the volume of the pentagonal prism formula.
Step 3: Write the numerical value of volume so obtained with an appropriate unit

How to Find the Height When the Volume of a Pentagonal Prism Given?

In case the volume of a pentagonal prism is given, together with the apothem and base, then we can find the height of the pentagon prism

Step 1: Identify the given values, the volume of the pentagonal prism, apothem length, and base length.
Step 2: Divide the volume (5/2abh) of the pentagonal prism by its base area (5/2ab).

How Do You Find the Volume of an Irregular Pentagonal Prism?

To find the volume of an irregular prism, just put the values of base area and height in the volume of prism formula, that is V = base area × height. So is the case with an irregular pentagonal prism.

Step 1: Check all the information that is given, base area and height
Step 2: Put the values in formula: V = base area × height

Note that the height of the pentagonal prism is the total height of the prism that is perpendicular from the vertex of the face to the base of the prism.