Volume of Pyramid Derivation

Let’s consider a rectangular pyramid and a prism where the base and height of both the pyramid and the prism are the same. Now take a rectangular pyramid full of water and pour the water into the empty prism. We can observe that only one-third part of a prism is full. Thus, we can say that volume of pyramid is 1/3 of the volume of prism.

Hence, the volume of a pyramid is equal to one-third of the volume of a prism if the base and height of both the pyramid and the prism are the same. So,

Volume of Pyramid = (1/3) × [Volume of Prism]

We know that,

Volume of Prism = A.H cubic units

Hence,

Volume of Pyramid (V) = (1/3) A.H cubic units

where,

• A is Base Area of Pyramid
• H is Height of Pyramid

Volume of Triangular Pyramid

Pyramid that has a triangular base is called the triangular pyramid. A triangular pyramid has three triangular faces and one triangular base, where the triangular base can be equilateral, isosceles, or a scalar triangle.

A triangular pyramid is also referred to as a tetrahedron. The formula for the volume of triangular pyramid is given,

Volume of Triangular Pyramid = 1/3 A.H cubic units 

We know that,

Area of Triangle(A) = 1/2 b × h

where

• b is Length of Base of Triangle
• h is Height of Base of Triangle

Now, volume of triangular pyramid (V)= 1/3 (1/2 b × h)H cubic units

V = 1/6 bhH cubic units

Hence,

Volume of Triangular Pyramid (V) = 1/6 b.h.H cubic units

where,

• b is Base of Triangular Base of Pyramid
• h is Height of Triangular Base of Pyramid
• H is Height of Pyramid

Volume of Square Pyramid

Pyramid that has a square base is called the square pyramid. A square pyramid has four triangular faces and one square base.

Formula for the volume of square pyramid is given,

Volume of Square Pyramid = 1/3 A.H cubic units

Area of Square = a²

Now, the volume of the square pyramid (V)= 1/3 (a²) H cubic units

V = (1/3) a²H cubic units

Hence, 

Volume of Square Pyramid (V) = (1/3) a²H cubic units

where,

• a is Side of Base Square
• H is height of Pyramid

Volume of Rectangular Pyramid

Pyramid that has a rectangular base is called the rectangular pyramid. A rectangular pyramid has four triangular faces and one rectangular base.

The formula for the volume of rectangular pyramid is given,

Volume of Rectangular Pyramid = 1/3 A.H cubic units

Area of Rectangle = l × w

Now, the volume of the rectangular pyramid (V)= 1/3 (l × w) H cubic units

V = 1/3 (l × w × H) cubic units

Hence,

Volume of Rectangular Pyramid (V)= 1/3 (l × w × H) cubic units

where,

• l is length of Base Rectangle
• w is width of Base Rectangle
• H is height of Pyramid

Volume of Pentagonal Pyramid

Pyramid that has a pentagonal base is called the pentagonal pyramid. A pentagonal pyramid has five triangular faces and one pentagonal base.

Formula for the volume of pentagonal pyramid is given,

Volume of Pentagonal Pyramid = 1/3 A.H cubic units

Area of Pentagon = (5/2) S × a

Now, the volume of the pentagonal pyramid (V)= 1/3 (5/2 S × a) H cubic units

V = 5/6 aSH cubic units

Hence,

Volume of Pentagonal Pyramid (V) = 5/6a.S.H cubic units

where,

• S is Length of Side of Pentagon Base
• a is Apothem Length of Side of Pentagon Base
• H is Height of Pyramid

Volume of Hexagonal Pyramid

Pyramid that has a hexagonal base is called the hexagonal pyramid. A hexagonal pyramid has six triangular faces and one hexagonal base.

Formula for the volume of the hexagonal pyramid is given,

Volume of Hexagonal Pyramid = 1/3 A.H cubic units

Area of Hexagon = 3√3/2 a²

Now, the volume of the hexagonal pyramid (V)= 1/3 (3√3/2 a²) H cubic units

V = √3/2 a² H cubic units

Hence,

Volume of Hexagonal Pyramid (V) = √3/2 a² H cubic units

• a is Edge of Side of Hexagon Base
• H is Height of Pyramid

Pyramid is a three-dimensional shape whose base is a polygon, and all its triangular faces join at a common point called the apex. The pyramids of Egypt are real-life examples of pyramids. Volume of a pyramid is the space occupied by that pyramid and is calculated by the formula, V = 1/3×(Area of Base)×(Height)

In this article, we have covered Pyramid definition, volume of the pyramid formula, derivation and others in detail.