What is the volume of a cone?

The volume of a cone is defined as the amount of space occupied by the hollow cone with the circular base. Cone is a 3-Dimensional structure having height, a circular base, and a taper vertex. The line which is adjoining the vertex of the cone is the height (h) of the cone and having a circular base is called the diameter (d) of the cone example – Happy birthday cap, pencil tip, etc.

A real-life example of a cone is when we go to the ice cream parlor, the ice cream cone is filled with an edible cream that covers the total amount of ice cream (up to the circular base) is the volume of the cone.

Get Maths Help Download DASA/CIWG E-BOOK

The volume of Cone Formula Derivation

A cone is defined as a hollow circular pyramid. A right cone is defined as the vertex is just over the center of the surface of a circular base.

The formula for the volume of a cone is:-

 The volume of the cone = ⅓ 𝜋r²h

Where,

      r =is the base of the circular radius

      h= is the height of the cone measured from the center of the cone.

The volume of a cone with a slant Height 

By applying Pythagoras theorem on the hollow cone with height (h), radius (r), and slant height is (l)

Now,

       l² = h² + r²

       h² = l² – r²

       h = √l²- r²

putting the value oh h = √l²- r²

Now, The Volume of the cone =  1/3𝜋r²( l²-r²)

Book a Free Mathematics Class

Derivation of the formula of Volume of the cone

The derivation of the cone can be understood by one simple activity taking one hollow empty cone and one empty hollow cylinder of the same height (h) and diameter (d). Now, start pouring water until the hollow cone is filled up to the brim. Now empty the hollow cone by pouring all the water into the hollow cylinder now you will at this stage the hollow cylinder is not filled fully.

Now repeat this activity one more time again you will see that the cylinder is not filled fully, Now repeat this activity for the third time now you will see that this time cylinder is filled.

Now we can say that volume of the cone is= ⅓ of the volume of the Cylinder

Now we put the value of Volume of Cylinder and cone then we get,

The volume of the cone = ⅓𝜋r²h

Ques 1. Find the volume of cone having h=3m ,r =2m. Use 𝜋 = 22/7 

Ans- we known the height the h= 3m and r= 2m

 Now put these value in formula,

     =1/3×22/7x(2²) x (3)

     =12.56m³

Ques 2. Find the volume of a cone having r= 3m, slant height (l) =5m.

Ans. We know, radius r=3m and slant height (l) = 5m

 Now we put the slant height formula to find the volume of the cone

 The Volume of the cone =  1/3𝜋r² (l²-r²)

 Now, put all the values in the above formula we get,

      =37.68m³