Volume of a Pyramid: Formula and Solved Example

Namrata Das logo

Namrata Das Exams Prep Master

Exams Prep Master

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.

Read More: Difference Between Relation and Function


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.

Pyramid
Pyramid

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:

Also Read:


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

Read More:

Value of Log 1 to 10

Area of Hollow Cylinder

Area of Segment of a Circle


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


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.

Square Pyramid
Square Pyramid

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.

Triangular Pyramid
Triangular Pyramid

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

Read More:

Factorial Formula

Centroid of a Triangle

Consistent system

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.

Pentagonal Pyramid
Pentagonal Pyramid

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

Rectangular Pyramid
Rectangular Pyramid

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

Hexagonal Pyramid
Hexagonal Pyramid

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.

Also read: 


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.

Read More:

Bias

Mode

Value of e


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?

Read More:


Sample Questions

Ques. Find the Volume of the square Pyramid if its base area is 100 unit sq. Its height is 24units. (2 marks)

Ans: Given,

Area of base = 100 unit sq.

Height = 24 units

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

therefore, 

Volume = 1/3 x 100 x 24

Volume = 800 unit cube

Hence, the Volume of the Pyramid is 800 units cube. 

Ques. If the Volume of a square pyramid is 240 units cube and the base area of the Pyramid is 60 units sq. Find the height of the Pyramid? (3 marks)

Ans: Given, The Volume of Pyramid = 240 units cube

The base area of the Pyramid = 60 units sq. 

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

Placing the values in the formula-

240 = 1/3 x 60 x height

240 = 20 x height

Height = 240 / 20

Height = 12 units

Hence, the height of the Pyramid is 12 units. 

Ques. What is the formula to find the volume of a regular pyramid? (1 mark)

Ans. The formula for the same is: The formula for Volume of Pyramid = 1/3 x base area x height

Ques. What is a Tetrahedron Pyramid? (2 marks)

Ans. A tetrahedron is just another name for a triangular pyramid. It is a particular type of triangular Pyramid in which all the faces, including the base of the Pyramid, are triangular. It is exclusively the only type of Pyramid where any of the Pyramid's sides can act as the base, and any vertices can act as the apex for the Pyramid. 

Ques. What will be the height of the isosceles triangular Pyramid that has a volume of 75 cubic feet and a base length of 12 feet? (3 marks)

Ans. Given,

The Volume of Pyramid = 75 cubic feet

Length of Pyramid = 12 feet

Volume of triangular pyramid = 1/6 x length x breadth x height

75 = 1/6 x 12 x 12 x height

75 = 24 x height

Height = 3.125 feet

Hence, the height of the Pyramid is 3.125 feet. 

Ques. Find the volume of a pyramid whose base is square. The sides of the base are 10 cm each and the height of the pyramid is 18 cm.

Ans. To find the volume of a pyramid, we will use the formula: V = 1/3 A H

As the base of the pyramid is a square, the area of the base will be a2 = 10 x 10 = 100 cm2

= 1/3 x 100 cm 2 x 18 cm

= 100 x 6

= 600 cm3.

Ques. Find the volume of a pyramid whose base is square, and the sides of the base are 12 cm each and the height is 21 cm. (2 marks)

Ans. To find the volume of a pyramid, we will use the formula: V = 1/3 A H

As the base of the pyramid is a square, the area of the base is a2 = 12 x 12 = 144 cm2

= ? x 144 cm2 x 21 cm

= 144 x 7

= 1008 cm3

Also check:

CBSE X Related Questions

  • 1.
    In a trapezium \(ABCD\), \(AB \parallel DC\) and its diagonals intersect at \(O\). Prove that \[ \frac{OA}{OC} = \frac{OB}{OD} \]


      • 2.
        Find the zeroes of the polynomial \(2x^2 + 7x + 5\) and verify the relationship between its zeroes and coefficients.


          • 3.
            Prove that \(\dfrac{\sin \theta}{1 + \cos \theta} + \dfrac{1 + \cos \theta}{\sin \theta} = 2\csc \theta\)


              • 4.

                Directions: In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R).
                Choose the correct option from the following:
                (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
                (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
                (C) Assertion (A) is true, but Reason (R) is false.
                (D) Assertion (A) is false, but Reason (R) is true.

                Assertion (A): For any two prime numbers $p$ and $q$, their HCF is 1 and LCM is $p + q$.
                Reason (R): For any two natural numbers, HCF × LCM = product of numbers.

                  • Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
                  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
                  • Assertion (A) is true, but Reason (R) is false.
                  • Assertion (A) is false, but Reason (R) is true.

                • 5.

                  In the adjoining figure, TS is a tangent to a circle with centre O. The value of $2x^\circ$ is

                    • 22.5
                    • 45
                    • 67.5
                    • 90

                  • 6.

                    From one face of a solid cube of side 14 cm, the largest possible cone is carved out. Find the volume and surface area of the remaining solid.
                    Use $\pi = \dfrac{22}{7}, \sqrt{5} = 2.2$

                      Comments


                      No Comments To Show