A cone is a three-dimensional figure with a circular base and apex or vertex. Sets of line segments join every point in the circular base to the apex. Another way to visualize a cone is as a pile of non-congruent circular discs stacked one on top of another. The diameters of adjacent discs are in constant ratio. The volume of a cone can be described as its capacity or the volume of space occupied by the cone. It can be represented mathematically as below:

r is the radius of the base, and the height of the cone is h.

A cone is a three-dimensional figure with a circular base and an apex/vertex. Line segments connect every point on the circular base to the vertex.A cone can be thought of as stacking non-congruent circular discs one upon the other such that there is a constant ratio of the diameters of the adjacent discs. The volume of the cone is its capacity or it is the measure of space occupied by the cone. Mathematically, it can be expressed as:

Where, r =base radius and h= height of the cone. 

Derivation: Activity to compute the volume of a cone. Take a conical flask and a cylindrical container equal in base radius and equal in height. Fill the flask up to the brim with water. Start pouring this water to the cylindrical container. You'll find that it doesn't fill up the cylinder completely. Fill the conical flask with water all over again to the brim. Pour this water into the cylinder, you'll see there is still empty space in the container. Try this experiment again. You'll notice that this time the cylindrical container is perfectly filled. We can conclude from this experiment that the volume of a cylinder is thrice the volume of a cone having the same height and base radius. That means the volume of a cone is one-third of the volume of a cylinder having the same height and base radius. Now, you know that the volume of a cylinder =

Hence volume of a cone=

Volume of Cone in Real Life: In our day-to-day life, we encounter various types of cones. We can use the volume of the cone formula in various productions and for calculating the volume or capacity of traffic cones, ice cream cones, funnels, and even regular cylinders. You can easily determine the volume of a cone if you have the measurements of its height and radius. 

Solved Examples: 

Q1: Find the volume of a cone, if radius is 4 cm and height is 9 cm. Solution: Radius r= 4 cm Height h= 9 cm By using the volume of a cone formula, Volume of cone =

Q2: Find the volume of a cone that has a base radius of 8 cm and slant height (l) of 13 cm.

Solution: 

Given, Radius r= 8 cm 

Slant height l = 13 cm 

Slant height,

Using formula of cone,

Volume of cone = 

Derivation

Here's an activity for calculating the volume of a cone.

Take a conical flask and a cylindrical container of the same base radius and the same height.

Keep pouring water to the flask till brimful. Start pouring this water into the cylindrical container. You will notice that it does not fill up the cylinder completely.

Repeat filling the conical flask completely with water. When you put this water into the cylinder, you will notice that there is still some empty space in the container.

Do this experiment yet once again. This time you will see that the cylindrical container is completely filled. 

We can conclude from this experiment that the volume of a cylinder is thrice the volume of a cone having the same height and base radius. This means that the volume of a cone is equal to one-third of the volume of the cylinder having the same height and base radius.

Now, you know that the volume of a cylinder =

Hence, 

Volume of cone=

Question 1: Find the volume of a cone, if the radius is 4 cm and height is 9 cm.

Solution:

Radius r = 4 cm

Height h = 9 cm

Using the volume of a cone formula,

Volume of cone =

Question 2: Find the volume of a cone that has a base radius of 8 cm and slant height (l) of 13 cm.

Solution:

Given,

Radius r = 8 cm

Slant height l = 13 cm

Slant height, 

Using formula, 

Volume of cone=

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