Pentagonal Pyramid - Definition, Properties, Formulas and Examples

What Is A Pentagonal Pyramid?

In Geometry, a pentagonal pyramid is a pyramid with a pentagonal base and five triangular faces that meet at a point known as the apex. The regular pentagonal pyramid has a base that is pentagonal in shape and lateral faces that are equilateral triangles. It is one of the Johnsons solid.

Pentagonal Pyramid

What are the Properties of a Pentagonal Pyramid?

The following are the properties of a pentagonal pyramid.

• It has 6 faces.

• The five side faces are triangular in shape.

• The base is a pentagon.

• It has 6 vertices.

• It has 10 edges.

• A pentagonal pyramid can also have an isosceles triangle (Triangle with two sides of equal length) as its lateral sides or lateral faces.

What is the Volume of the Pentagonal Pyramid?

The volume of a pentagonal pyramid is defined as the total space occupied by the pyramid in the three-dimensional space. The volume of a pentagonal pyramid is measured in cubic units.

The volume of any pyramid is calculated by multiplying the area of its base by its height and dividing the product by 3. Therefore, the volume of pyramid is given as,

Volume: 13 Base Area Height

As pentagonal pyramid has pentagonal base, therefore, the area of pentagon is calculated as:

Area = 1.72l²

Here, l is the length of one side of a pentagon.

The above formula for the area of the pentagon is calculated by dividing the pentagon into 5 triangles and finding the area of each triangle separately.

Therefore, the volume of pentagonal pyramid using the above expression for the area of given pentagon is calculated as:

Volume =1.72 3l²h

Here, l is the length of the one side of a pentagonal base and h is the height of the pyramid.

Volume of pentagonal pyramid

Let us understand the volume of the pentagonal pyramid with an example.

Example:

Find the volume of the pyramid with a pentagonal base 1 m and height 3 m.

Solution:

Pentagonal base : 1 m

Height of pentagon: 3 m

Now, we will use the volume of pentagonal pyramid formula with these values:

V = 1.72 31² 3

V = 1.72 3 3

V = 1.72

Hence, the volume of pentagonal pyramid with base length (1 m) and height (3 m) is given as 1.72m³.

What is the Surface Area of the Pentagonal Pyramid?

The surface area of a pentagonal pyramid is the entire surface occupied by the pyramid in three-dimensional space. The surface area of a pentagonal pyramid is measured in square units.

The surface area of any pyramid is calculated by adding the area of all the faces of the pyramid. Pentagonal pyramid has one pentagonal base and 5 triangular faces, and we used the following formula to calculate the area of the pentagonal face.

Area = 1.72l²

Here, l is the length of one side of the pentagonal base.

On the other hand, the area of triangle faces of pyramid is calculated by using the formula for area of triangle:

A = 12 Base Height

The five triangular faces of the pentagonal pyramid are congruent ( Congruent triangles have the same shape and same size).. Accordingly, the surface area of pentagonal pyramid is calculated as:

Surface Area of Pentagonal Pyramid Formula (AS) : 1.72l² + 52bh

Surface area of pentagonal pyramid

Let us understand with an example.

Example:

Find the surface area of a pentagonal pyramid with height 5 m and side length 1 m?

Solution:

Height = 5 m

Length = 1 m

Now, we will use these values in the surface area of pentagonal pyramid formula:

(AS) : 1.72l² + 52bh

(AS) : 1.72(1)² + 52(1)(5)

(AS) : 1.72(1)² + 12.5

(AS): 14.22

Therefore, the surface area of the pentagonal pyramid is equal to 14.22 m².

In short, a three-dimensional shape with a pentagonal base and five triangular faces meeting is termed the pentagonal pyramid. As you have understood what a pentagonal pyramid is. Now, you can easily solve and practice sums based on the volume and surface area of the pentagonal pyramid using the formula discussed above.