Volume of a Pyramid: Formula and Solved Example

A pyramid is a 3-dimensional closed polygon that has a polygon base and triangular faces, all connecting at the top. The Pyramid base can be of any shape like an equilateral triangle (a triangle with all equal sides), a square, or a Pentagon, etc. A pyramid is a typical shape that connects all the polygon sides from the base to the top at a common point or apex, giving it its final shape—ever wondered how you would find the Volume of such a figure. Let's learn the formula to find the Volume of a Pyramid and discuss some important questions.

Key Takeaways: Pyramid, Volume of Pyramid, the formula for the Volume of Pyramid, practice questions on volume of Pyramid.

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What is a Pyramid?

A pyramid is a closed three-dimensional figure. The base of the polygon can be of various shapes, but all the faces of the polygon connect at the top of the 3-D figure to complete the structure of the Pyramid. It is interesting to know that all the sides of a pyramid are always triangular. In mathematical language, these triangular sides of the Pyramid are also known as faces, and the top connecting point of all the faces is the apex. The complete structure is obtained only after connecting all the faces to the apex at the top. In some cases, these triangular faces are also referred to as lateral faces to distinguish them from the Pyramid base.

The faces or the sides of the Pyramid are also known as the Lateral Faces. The base of the Pyramid can be square, triangle, pentagon, or any closed polygon. The top of the Pyramid is known as the apex.

The video below explains this:

Volume of a Pyramid Detailed Video Explanation:

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What is the Volume of a Pyramid?

The Volume of the Pyramid defines the total space enclosed in between all the faces of the Pyramid, i.e., the total space inside the closed Pyramid. Each of the pyramids has a different volume formula depending upon the shape of the Pyramid's base. The Volume of the Pyramid is measured in cubic units like any other volume unit.

The formula used to calculate the Volume of the Pyramid is :

The Volume of Pyramid = 1/3 x base area x height

Where, V = Volume, A = Area and H = height

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Derivation of Volume of a Pyramid

In order to find the volume of a pyramid, we have to know the total capacity of the given pyramid. The formula for the pyramid’s volume is represented by one-third of the product of the area of the base to its height. 

Its volume is measured in the following units:

• in³
• ft³
• cm³
• m³ etc

Read More: Volume of a Pyramid Formula & Solved Example

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Types of Pyramid

A Pyramid is divided into various types depending upon the base shape of the Pyramid. So, let us now see the various types of Pyramid:

Square Pyramid

A square pyramid is a pyramid having a square base. It consists of one square base and four triangular faces. It has eight edges, five vertices, and four faces.

The area of a square is given by:

A = a²

Where a represents the length of the side of the square.

Thus, the volume of a square-sided pyramid is;

V = 1/3 x Area of square base x Height

V = 1/3 x a² x H

V = 1/3 a² H

Triangular Pyramid

A triangular pyramid is a pyramid having a triangle base. This base triangle can be an equilateral, isosceles triangle, or a scalar triangle. It has a triangular base, four faces, six edges, and four vertices. A triangular pyramid is also known as a tetrahedron.

As we know, the area of a triangle:

A = 1/2 b x h

where b represents the base of the triangle and h is the altitude.

Hence, the volume of a triangular pyramid;

V = 1/3 x Area of triangular base x Height of pyramid

V = 1/3 x (1/2 bh) H

V = 1/6 bhH

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Pentagonal Pyramid

Pentagonal Pyramid is a pyramid having a pentagonal shape. The lateral faces of this Pyramid as well are triangular. The Pentagonal pyramid has six vertices, six faces, and ten edges.

Rectangular Pyramid

A rectangular pyramid consists of a base in a rectangular shape. As, we know that the area of the rectangle is equal to the product of its length and width, such as;

A = Length x Width

A = lw

Therefore, the volume of a rectangular pyramid is given by;

V = 1/3 x A x H

V = 1/3 lwH

Hexagonal Pyramid

The volume of a hexagonal pyramid, whose base is a regular hexagon shape is represented by:

Volume = 1/3 x area of base x height

V= 1/3 x 3√3/2 a² x H

V = √3/2 a² H

Where a = the side length of the hexagon base and, 

H = the height of the pyramid.

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Properties of Pyramid

A pyramid has various features and properties that help you identify it as a pyramid. Some of the properties of a pyramid are listed below:

1. A Pyramid is a closed 3-dimensional figure.
2. It has three parts- Base, Face, and Apex.
3. The base of the Pyramid is always a closed polygon.
4. All the flat faces of the Pyramid except the base are also known as the Lateral Faces of the Pyramid.
5. The line segments which intersect faces are known as the edges of the Pyramid.
6. The top point at which three or more edges meet is known as the vertex.
7. Apex is the point opposite to the base where all the faces join in the Pyramid.

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Things to Remember

• The Pyramid is a closed polygon whose all sides connect at the top apex to give it the defined shape. The Volume of the Pyramid is defined as the number of cubic units covered by the pyramidal shape. 
• As we know, the pyramid base can have varied shapes. Hence the Volume of the Pyramid depends on its particular shape. The formula to find the Volume of the Pyramid is: The Volume of Square Pyramid = 1/3 x base x height
• Volume of triangular pyramid = 1/6 x length x breadth x height
• Volume of pentagonal pyramid = 5/6 length x breadth x height
• Volume of hexagonal pyramid = length x breadth x height
• The volume of a rectangular pyramid is given by: V = 1/3 lwH?

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