Volume of a Pyramid Formula

In geometry, a pyramid is a three-dimensional shape whose base is a polygon, and all its triangular faces join at a common point called the apex. A pyramid is formed by joining all the sides of a polygon base to an apex. The pyramids of Egypt are real-life examples of pyramids. There are different types of pyramids based on the shape of the base of the pyramid. The different types of pyramids are triangular pyramids, square pyramids, rectangular pyramids, pentagonal pyramids, etc. The side faces of any type of pyramid are triangular, and one side of each triangular face merges with the side of the base. The top of the pyramid is called the apex, and the side faces are called the lateral faces of a pyramid.

The volume of a Pyramid

The volume of a pyramid refers to the total space enclosed between all the faces of a pyramid; in simple words, the total space inside a closed pyramid. The formula for the volume of a pyramid is equal to one-third of the product of the base area and the height of the pyramid and is usually represented by the letter “V”.

The formula for the volume of a pyramid is given as follows,

The volume of a Pyramid = 1/3 × base area × height

V = 1/3 AH cubic units

Where V is the volume of the pyramid,

A is the base area of the pyramid and

H is the height or altitude of a pyramid.

Derivation of the volume of a pyramid

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. So, repeat the experiment once more, and we will notice that there is still some vacant space in the prism. Once again, repeat the experiment, and this time we can notice that the prism is filled up to the brim. 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,

The volume of a prism = 3 × [Volume of a pyramid]

i.e.,

The volume of a pyramid = (1/3) × [Volume of a prism]

We know that,

The volume of a prism = AH cubic units

Hence,

Volume of a pyramid (V) = (1/3) AH cubic units

A is the base area of the pyramid and H is the height or altitude of a pyramid. 

The volume of a triangular pyramid

The 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 the triangular pyramid is given,

The volume of the triangular pyramid = 1/3 AH cubic units 

Where H is the height of the pyramid and A is the area of the base

We know that,

Area of a triangle = 1/2 b × h

Where b is the length of the base of the triangle and h is its height.

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

V = 1/6 bhH cubic units

Hence,

 The volume of the triangular pyramid (V)= 1/6 bhH cubic units

The volume of a square pyramid

The pyramid that has a square base is called the square pyramid. A square pyramid has four triangular faces and one square base. The formula for the volume of the square pyramid is given,

The volume of the square pyramid = 1/3 AH cubic units

Where H is the height of the pyramid and A is the area of the base

Area of a square = a²

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

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

V = (1/3) a²H cubic units

Hence, 

The volume of the square pyramid (V)= (1/3) a²H cubic units

The volume of a rectangular pyramid

The 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 the rectangular pyramid is given,

The volume of the rectangular pyramid = 1/3 AH cubic units

Where H is the height of the pyramid and A is the area of the base

Area of a rectangle = l × w

Where l is the length of the rectangle and w is its width.

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

V = 1/3 (l × w × H) cubic units

Hence,

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

The volume of a pentagonal pyramid

The pyramid that has a pentagonal base is called the pentagonal pyramid. A pentagonal pyramid has five triangular faces and one pentagonal base. The formula for the volume of the pentagonal pyramid is given,

The volume of the pentagonal pyramid = 1/3 AH cubic units

Where H is the height of the pyramid and A is the area of the base

Area of a pentagon = (5/2) S × a

Where S is the length of the side of a pentagon and a is its apothem length.

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

V = 5/6 aSH cubic units

Hence,

The volume of the pentagonal pyramid (V)= 5/6 aSH cubic units

The volume of a hexagonal pyramid

The pyramid that has a hexagonal base is called the hexagonal pyramid. A hexagonal pyramid has six triangular faces and one hexagonal base. The formula for the volume of the hexagonal pyramid is given,

The volume of the hexagonal pyramid = 1/3 AH cubic units

Where H is the height of the pyramid and A is the area of the base

Area of a hexagon = 3√3/2 a²

Where a is the length of the side of the hexagon.

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,

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

Sample Problems

Problem 1: What is the volume of a square pyramid if the sides of a base are 6 cm each and the height of the pyramid is 10 cm?

Solution:

Given data,

Length of the side of the base of a square pyramid = 6 cm

Height of the pyramid = 10 cm.

The volume of a square pyramid (V) = 1/3 × Area of square base × Height

Area of square base = a² = 6² = 36 cm²

V = 1/3 × (36) ×10 = 120 cm³

Hence, the volume of the given square pyramid is 120 cm³.

Problem 2: What is the volume of a triangular pyramid whose base area and height are 120 cm²  and 13 cm, respectively?

Solution:

Given data,

Area of the triangular base = 120 cm²

Height of the pyramid = 13 cm

The volume of a triangular pyramid (V) = 1/3 × Area of triangular base × Height

V = 1/3 × 120 × 13 = 520 cm³

Hence, the volume of the given triangular pyramid = is 520 cm³

Problem 3: What is the volume of a triangular pyramid if the length of the base and altitude of the triangular base are 3 cm and 4.5 cm, respectively, and the height of the pyramid is 8 cm?

Solution:

Given data,

Height of the pyramid = 8 cm

Length of the base of the triangular base = 3 cm

Length of the altitude of the triangular base = 4.5 cm

Area of the triangular base (A) = 1/2 b × h = 1/2 × 3 × 4.5 = 6.75‬ cm²

The volume of a triangular pyramid (V) = 1/3 × A × H

V = 1/3 × 6.75 × 8 = 18 cm³

Hence, the volume of the given triangular pyramid is 18 cm³

Problem 4: What is the volume of a rectangular pyramid if the length and width of the rectangular base are 8 cm and 5 cm, respectively, and the height of the pyramid is 14 cm?

Solution:

Given data,

Height of the pyramid = 14 cm

Length of the rectangular base (l) = 8 cm

Width of the rectangular base (w) = 5 cm

Area of the rectangular base (A) = l‬ × w = 8 × 5 = 40 cm²

We have,

The volume of a rectangular pyramid (V) = 1/3 × A × H

V = 1/3 × 40 × 14 = 560/3 = 186.67 cm³

Hence, the volume of the given rectangular pyramid is 186.67 cm³.

Problem 5: What is the volume of a hexagonal pyramid if the sides of a base are 8 cm each and the height of the pyramid is 15 cm?

Solution:

Given data,

Height of the pyramid = 15 cm

Length of the side of the base of a hexagonal pyramid = 6 cm

Area of the hexagonal base (A) = 3√3/2 a² = 3√3/2 (6)² = 54√3 cm²

The volume of a hexagonal pyramid (V) = 1/3 × A × H

V = 1/3 × 54√3 × 15 = 270√3 cm³

Hence, the volume of the given hexagonal pyramid is  270√3 cm³.

Problem 6: What is the volume of a pentagonal pyramid if the base area is 150 cm² and the height of the pyramid is 11 cm?

Solution:

Given data,

Area of the pentagonal base = 150 cm²

Height of the pyramid = 11 cm

The volume of a pentagonal pyramid (V) = 1/3 × Area of pentagonal base × Height

V = 1/3 × 150 × 11 = 550 cm³

Hence, the volume of the given pentagonal pyramid = 550 cm³