 # Volume of a Pyramid

The volume of a pyramid depends upon the type of pyramid’s base, whether it is a triangle, square or rectangle. A pyramid is a polyhedron figure which has only one base. The base of the pyramid is a poly sided figure. Hence, the formula to find not only volume but also the surface area of a pyramid will be based on the structure of its base and height of the pyramid. Now let us find out the formula for the triangular pyramid, square pyramid and rectangular pyramid in this article.

## Volume of a Pyramid Formula and Derivation

To find the volume of a pyramid, we need to know the total capacity of the given pyramid. The formula for the pyramid’s volume is given by one-third of the product of the area of the base to its height. Such as:

Volume = 1/3 x Area of the Base x Height

 V = ⅓ A × H

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

It’s volume is measured in following units:

• in3
• ft3
• cm3
• m3 etc

Note: Always make sure to keep all the measurement units same while calculating the volume.

Now let us find out the formula for the volume of different types of pyramids.

### Volume of Triangular Pyramid

A triangular pyramid has base in triangle shape. As we know, the area of a triangle;

A = 1/2 b x h

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

Therefore, 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

### Volume of Square Pyramid

A square-based pyramid has base in square shape. As we know, the area of a square is given by;

A = a2

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

Hence, 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

### Volume of Rectangular Pyramid

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

A = Length x Width

A = lw

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

V = 1/3 x A x H

V = 1/3 lwH

### Examples

Question 1: What is the volume of a pyramid whose base is square in shape? The sides of the base are 10 cm each and the height of the pyramid is 18cm.

Solution: To find the volume of a pyramid, We will use the formula – V = ⅓ A H

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

= ⅓ x 100 cm 2 x 18 cm

= 100 x 6

= 600 cm3

Question 2: What is the volume of a pyramid whose base is square in shape? The sides of the base are 12 cm each and the height is 21cm.

Solution: To find the volume of a pyramid, We will use the formula – V = ⅓ A H

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

= ⅓ x 144 cm 2 x 21 cm

= 144 x 7

= 1008 cm3

Question 3: If the base of the pyramid is rectangular in shape having length is 7cm and the width is 5 cm and the height of the pyramid is 10cm, then find its volume.

Solution: Given, length of rectangular base of pyramid = 7cm

width = 5cm

and height of the pyramid = 10cm

We know, the volume of rectangular pyramid,

V = 1/3 l w H

Substituting the values we get;

V = 1/3 x 7 x 5 x 10

V = 350/3 = 116.66 cubic.cm.

Question 4: For triangular pyramid, the area of the base is 135 cubic cm. Find its volume if the height of pyramid is 7cm.

Solution: Given the area of the base of pyramid = 135 cu.cm

Volume of Pyramid = 1/3 x A x H

V = 1/3 x 135 x 7

V = 45 x 7

V = 315 Cu.Cm.

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