Volume of a Pyramid Formula

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.

A 3-D shape with a polygonal base and triangular faces that meet at a common point apex is called a Pyramid. the image for the same is added below:

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.

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”.

Formula for the volume of a pyramid is given as follows,

Volume of a Pyramid = 1/3 × Base Area × Height

V = 1/3 A.H cubic units

where,

• V is Volume of Pyramid
• A is Base Area of Pyramid
• H is Height 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. 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

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

• Length of Side of Base of Square Pyramid = 6 cm
• Height of Pyramid = 10 cm

Volume of 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, 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

• Area of Triangular Base = 120 cm²
• Height of Pyramid = 13 cm

Volume of a Triangular Pyramid (V) = 1/3 × Area of Triangular Base × Height

V = 1/3 × 120 × 13 = 520 cm³

Hence, 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

• Height of Pyramid = 8 cm
• Length of Base of Triangular Base = 3 cm
• Length of Altitude of Triangular Base = 4.5 cm

Area of Triangular Base (A) = 1/2 b × h = 1/2 × 3 × 4.5 = 6.75‬ cm²

Volume of Triangular Pyramid (V) = 1/3 × A × H

V = 1/3 × 6.75 × 8 = 18 cm³

Hence, 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

• Height of Pyramid = 14 cm
• Length of Rectangular Base (l) = 8 cm
• Width of Rectangular Base (w) = 5 cm

Area of Rectangular Base (A) = l‬ × w = 8 × 5 = 40 cm²

We have,

Volume of Rectangular Pyramid (V) = 1/3 × A × H

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

Hence, 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

• Height of Pyramid = 15 cm
• Length of Side of Base of Hexagonal Pyramid = 6 cm

Area of Hxagonal Base (A) = 3√3/2 a² = 3√3/2 (6)² = 54√3 cm²

Volume of Hexagonal Pyramid (V) = 1/3 × A × H

V = 1/3 × 54√3 × 15 = 270√3 cm³

Hence, 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

• Area of Pentagonal Base = 150 cm²
• Height of Pyramid = 11 cm

Volume of Pentagonal Pyramid (V) = 1/3 × Area of Pentagonal Base × Height

V = 1/3 × 150 × 11 = 550 cm³

Hence, volume of the given pentagonal pyramid = 550 cm³

What is a Pyramid?

Pyramid is a three-dimensional geometric shape with a polygonal base and triangular faces converging at a common point, called the apex.

What is the formula for volume of pyramids?

Formula for volume of pyramid is Volume = 1/3 × Base Area × Height

What is the volume of a triangular pyramid?

Volume of Triangular Pyramid is found the formula, Volume = 1/3 × Aarea of Base × Height = 1/6 b.h.H cubic units

What is the volume of the square pyramid?

Volume of Square Pyramid is found the formula, Volume = 1/3 × Aarea of Base × Height = (1/3) a²H cubic units