Volume of Cone

Cone

A cone is defined as a distinctive three-dimensional geometric figure with a flat and curved surface pointed toward the top.

The term “cone” is derived from the Greek word “konos”, which means a wedge or a peak.
The pointed end is the apex (vertex), whereas the flat surface is called the base.

Properties of Cone

It has 1 curved surface, 1 flat face and 1 vertex.
It has no edges.

Types of Cone

There are two types of cones:

1. Right Circular Cone: A cone whose axis is perpendicular to the base.

2. Oblique Cone: A cone whose axis is not perpendicular to the base.

Volume of a Cone

The volume of a cone is described as the amount of space or capacity a cone takes up.
Its volume is expressed in cubic units such as cm³, m³, in³ etc.
A cone's volume can alternatively be measured in litres.

Formula for Volume of a Cone

1. Volume of a Cone with Height and Radius

The formula for the volume of a cone is:

V = (1/3)πr²h

Where,
r is the radius of the base of the cone
h is the height of the cone
π (pi) is a mathematical constant whose value is 22/7 or 3.14 approximately.

NOTE: The volume of a regular cone or right circular cone and the oblique cone can be calculated using the same formula.

2. Volume of a Cone with Slant Height

The slant height is the distance from the vertex or apex to the point on the outer line of the cone’s circular base.

By the Pythagoras Theorem

We know, h² + r² = l²
l = √ (r² + h²)
h = √ (l² - r²)

where:
h is the height of the cone,
r is the radius of the base, and
l is the slant height of the cone.

The volume of the cone in terms of slant height can be given as
V = (1/3)πr²h
= (1/3)πr²√(l² - r²)