Volumes of 3-D Figures - Pyramids Formulas | List of Volumes of 3-D figures - Pyramids Formulas You Should Know - BYJUS

# Volumes of 3-D Figures - Pyramids Formulas

Volume is the amount of space that an object occupies. A pyramid is a three-dimensional figure with a polygon-shaped base and triangular lateral faces meeting at a common point. Here we will focus on the formula used to calculate the volume of pyramids....Read MoreRead Less

### Introduction

Pyramids are three-dimensional structures classified on the basis of the shape of their base. A pyramid having a triangular base is known as a triangular pyramid. Similarly, a pyramid with a square base is called a square pyramid, and so on.

The lateral faces of a pyramid meet at a common point, known as the vertex.

The height of a pyramid is the perpendicular distance from its apex to its base.

 Type Diagram Properties Square pyramid 5 faces (1 square base and 4 triangular lateral faces)8 edges5 vertices Triangular pyramid 4 faces (1 triangular base and 3 triangular lateral faces)6 edges4 vertices

### List of Formulas

Here is the formula for calculating the volume of a pyramid:

The volume of a pyramid, v = $$\frac{1}{3} \times \text{B} \times \text{h}$$

Where,

• B is the area of the base.
• h is the height of the pyramid.

Hence, the volume of a pyramid is one-third of the product of its base area and its height.

### How do we use the Volume of a Pyramid formula?

To calculate the volume of a pyramid, we need to measure its base area and height. The base area can be calculated by using the area of a polygon formula.

The formulas for the volume of various types of pyramids, including the triangular pyramid, the square pyramid, and the rectangular pyramid, are shown here.

### Solved Examples

Example 1: A rectangular pyramid has a height of $$\frac{27}{4}$$ in. If its base area is 12 sq. in, what is its volume?

Solution:

According to the given data, height = $$\frac{27}{4}$$in

Base area = 12 sq.in

The volume of a pyramid, V = $$\frac{1}{3}\times \text{Area of the base} \times \text{height}$$

V = $$\frac{1}{3}\times 12 \times \frac{27}{4}$$  [Substitute values]

V = $$\frac{1 \times 12 \times 27}{3 \times 4}$$

V = $$\frac{1 \times 12 \times 27}{12}$$          [Simplify]

V = 27                   [Simplify]

Therefore, the volume of the rectangular pyramid is 27 cubic inches.

Example 2: Find the height of the pyramid given below that has a  volume of 36 cubic meters.

Solution:

According to the given data, volume = 36 cu.m

Base area = $$6 \times 4$$ = 24 sq.m

The volume of a pyramid = $$\frac{1}{3}\times \text{Area of the base} \times \text{height}$$

$$36=\frac{1}{3}\times 24 \times x$$  [Substitute values]

$$36=8 \times x$$            [Simplify]

$$\frac{36}{8}=x$$                  [Divide both sides by 8]

$$4.5=x$$                 [Simplify]

Example 3: Find the side length of the base of the pyramid given below that has a volume of 605 cubic centimeters.

Solution:

According to the given data, volume = 605 cu.cm

Base area = $$x \times x = x^2$$ sq.cm     [Area of square formula]

The volume of a pyramid = $$\frac{1}{3}\times \text{Area of the base} \times \text{height}$$

$$\frac{1}{3}\times x^2 \times 15=605$$      [Substitute values]

$$5 \times x^2=605$$               [Simplify]

$$x^2=\frac{605}{5}$$                      [Divide both sides by 5]

$$x^2=121$$

$$x=\sqrt{121}$$

$$x=11$$ cm                     [square root of 121 is 11]

Therefore, the side length of the base of the pyramid is 11 cm.

Example 4: The Great Pyramid of Giza is a pyramid having a square base. The base has a length of 440 cubits on each side. The pyramid stands at a height of 280 cubits. Calculate the volume of the Great Pyramid of Giza.

Solution:

According to the given data, the side length of the base = 440 cubits

Height of the pyramid = 280 cubits

Area of base = $$440 \times 440$$                                  [Area of square formula]

= 1,93,600 sq.cubits

Volume, V = $$\frac{1}{3}\times \text{Area of the base} \times \text{height}$$     [The volume of pyramid formula]

V = $$\frac{1}{3}\times \text{1,93,600} \times \text{280}$$                                     [Substitute values]

V = $$\frac{1}{3}\times \text{54,208,000}$$                                            [Multiply]

V= $$18,069,333.3~ \text{cubit}^3$$                                     [Simplify]

Therefore, the volume of The Great Pyramid of Giza is 18,069,333.3 cubic cubits.