# Volume of a Cone

## Volume of a Cone

Volume of a cone is given by the one third of the product of its area of base and height i.e. volume of cone =  πr2h/3. This formula can be understood and derived by the following activity:
1.     Take a hollow cylindrical jar of radius r and height h, whose volume is πr2h.
2.     Take a hollow cone which has the same height h and same radius r.
3.     Fill the cone with water and pour into the jar. And repeat it.

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In this activity, the cylindrical jar will be full filled with water when 3 cones of full water is pour into it. It shows that,

3 × Volume of cone = Volumeof the cylinder
or,       Volume of cone = 1/3 × Volume of the cylinder
= 1/3 × πr2h
= πr2h/3

Volume of a cone = πr2h/3

### Workout Examples

Example 1: Find the volume of a cone whose radius = 4 cm and height = 21 cm.
Solution: Here,
Height (h) = 21 cm

We know,
Volume of a cone = πr2h/3
= 1/3 × 22/7 × 42 × 21
= 352 cm3

Thus, the volume of the cone is 352 cm3.

Example 2: The vertical height of a right circular cone is three times its diameter and its volume is 54π cm3. Find its height.

Solution: Here,
Let, the diameter of cone be d.     d = 2r
Then, height (h) = 3d = 3 × 2r = 6r
Given, volume of cone = 54π cm3
i.e.      πr2h/3 = 54π
or,      r2h/3 = 54
or,      r2 × 6r = 162
or,      6r3 = 162
or,      r3 = 162/6
or,      r3 = 27
or,      r3 = 33
or,      r = 3 cm

Thus, height (h) = 6r = 6 × 3 cm = 18 cm.

Example 3: Calculate the volume of adjoining solid given in the figure.
Solution: Here,
The given figure is a combined solid of two cones.
Radius of both the cones (r) = 42 cm
Total height of the solid = 180 cm
Let the height of upper cone be h1 and the height of lower cone be h2.
So, h1 + h2 = 180 cm

We know,
Volume of the solid = V(upper cone) + V(lower cone)
= πr2h1/3 + πr2h2/3
= πr2(h1 + h2)/3
= 1/3 × 22/7 × 422 × 180     [  h1 + h2 = 180 cm]
= 332640 cm3

Thus, the volume of the solid is 332640 cm3.

You can comment your questions or problems regarding the volume of a cone here.