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

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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

square_pyramid

  • 5 faces (1 square base and 4 triangular lateral faces)
  • 8 edges
  • 5 vertices

Triangular pyramid

triangular_pyramid

  • 4 faces (1 triangular base and 3 triangular lateral faces)
  • 6 edges
  • 4 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.

 

pyramid_table

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?

 

pyramid_eg1

 

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.

 

pyramid_eg2

 

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.

 

pyramid_eg3

 

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.

Frequently Asked Questions

Three congruent pyramids can be formed from a cube of unit length. As a result, the volume of a pyramid is 1/3 times that of a cube. As a result, we have a 1/3 in the formula for the volume of a pyramid.

The distance from the apex along the lateral face of a pyramid is known as its slant height.

If the base of a pyramid is in the shape of a regular polygon, it is called a regular pyramid. However, if the base is in the shape of an irregular polygon, it is known as an irregular pyramid.

If the apex of a pyramid is not over the center of its base, it is known as an oblique pyramid.

A cubit is an ancient unit used to measure length. 1 cubit is generally equal to 18 inches.