Volume of a Cone - Calculator, Definition and Formula

Definition

A cone is a three-dimensional geometrical figure having a circular base that tapers smoothly to a point known as the vertex or apex. It has a similar cross-section as that of a pyramid. However, unlike a pyramid, the cone has a circular base, therefore, known as a circular cone.

There are four types of cones-

Right circular cone: – In a right circular cone, the angle between the lateral surface and the base is the same. The apex is positioned right above the base’s center point.

Oblique cone: – In a tilted cone, the angle between the lateral surface and the base is not the same all around the cone. The apex is not positioned right above the base’s center point.

Parabolic cone: – The intersection of a right circular cone and a plane forms a parabolic cone. All parabolas consist of a focus, an axis of symmetry, and a directrix. Similarly, a parabolic cone too has three points.

Elliptical cone: – It is a two-degree cone having the directrix in the form of an ellipse.

Formula

The volume of a cone is the space or capacity inside the cone. 

Think – Did you wonder why ice-cream cones are cones and not cylinders or any other shapes? 

The volume of a cone = (1/3) πr2h units3

Where ‘r’ is the radius of the base of the cone

‘h’ is the height of the cone 

Using the Pythagoras theorem, we can find the height ‘h’ of the right circular cone from the slant height ‘l’ of a cone. 

l2 = h2 + r2 

The volume of the cone is equal to the one-third volume of the cylinder. 

Proof – 

We know that the volume of a right circular cylinder is πr2h. -------------------------------------- (1)

where, 

r = radius of the circular cylinder 

h = height of the cylinder 

Also, the volume of the cone is (1/3) πr2h. ---------------------------------------------- (2) 

Equating (1) and (2), we get 

πr2h = (1/3) πr2h

Hence, it proved that the volume of a cylinder is three times the volume of the cone. 

Take a cylinder and three cones of the same height and circular base. Fill the cylinder with ice cream till its brim. Now transfer the ice cream from the cylinder to the cones. You will see all three cones filled with ice cream. That is a practical example of why ice cream sellers use cones to sell ice-creams rather than selling them in cylindrical tubs. Lesser the volume in a cone, the more the profit. Also, the shape of a cone provides a better grip to handle it. 

Try yourself: – Repeat the experiment with water. You will notice some leftover space in case of liquids. Repeat till the cones get exactly filled up to the brim.