FINDING THE VOLUME OF A CONE

We can find the volume of a sphere using the volume of a cylinder.

Cylinder is a solid which has a circular base.

We know the fact that the volume of any solid is equal to the product of base area and height of the solid.

So, the volume of a right circular cylinder of base radius ‘r’ and height ‘h’ is given by

V = (Base Area) x (Height)

The base of a cylinder is a circle, so for a cylinder,

Base Area = πr²

Therefore, 

Volume of a cylinder = πr²h cubic units

Consider a right circular cylinder and three right circular cones of same base radius and height as that of the cylinder.

The contents of three cones will exactly occupy the cylinder.

Then,

When we model the volume of a sphere, we will be getting the following result. 

3 x (Volume of a cone) = Volume of cylinder

3 x (Volume of a cone) = πr²h

Volume of the cone = 1/3 · πr²h cubic units

Example 1 :

Find the volume of the cone given below. Round your answer to the nearest tenth if necessary. Use the approximate of value of π, that is 3.14. 

Solution :

Step 1 :

Write the formula to find volume of a cone.

V = 1/3 · πr²h

Step 2 :

Substitute the given measures.

V ≈ 1/3 · 3.14 · 2² · 8

Simplify.

V ≈ 1/3 · 3.14 · 4 · 8

V ≈ 33.5

So, the volume of the given cone is about 33.5 cubic inches.

Example 2 :

Find the volume of the cone given below. Round your answer to the nearest tenth if necessary. Use the approximate of value of π, that is 3.14.

Solution :

Step 1 :

Write the formula to find volume of a cone.

V = 1/3 · πr²h -----(1)

Step 2 :

To find the volume, we need the radius of the cone. But, the diameter is given, that is 8 ft. So, find the radius. 

r = diameter / 2

r = 8/2

r = 4

Step 3 :

Substitute π ≈ 3.14, r = 4 and h = 9 in (1).

V ≈ 1/3 · 3.14 · 4² · 9

Simplify.

V ≈ 1/3 · 3.14 · 16 · 9

V ≈ 150.7

So, the volume of the given cone is about 150.7 cubic feet.

Example 3 :

Find the volume of the cone given below. Round your answer to the nearest tenth if necessary. Use the approximate of value of π, that is 3.14. 

Solution :

Step 1 :

Write the formula to find volume of a cone.

V = 1/3 · πr²h -----(1)

Step 2 :

To find the volume, we need the radius of the cone. But, the diameter is given, that is 15 cm. So, find the radius. 

r = diameter / 2

r = 15/2

r = 7.5

Step 3 :

Substitute π ≈ 3.14, r = 7.5 and h = 16 in (1).

V ≈ 1/3 · 3.14 · 7.5² · 16

Simplify.

V ≈ 1/3 · 3.14 · 56.25 · 16

V ≈ 942

So, the volume of the given cone is about 942 cubic cm. 

Example 4 :

Find the volume of the cone given below. Round your answer to the nearest tenth if necessary. Use the approximate of value of π, that is 3.14. 

Solution :

Step 1 :

Write the formula to find volume of a cone.

V = 1/3 · πr²h

Step 2 :

Substitute the given measures. 

V ≈ 1/3 · 3.14 · 2² · 3

Simplify.

V ≈ 1/3 · 3.14 · 4 · 3

V ≈ 12.6

So, the volume of the given cone is about 12.6 cubic feet.

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