There is an important field of study in mathematics, called **mensuration. **It is the study of areas and volumes two-dimensional as well as three-dimensional shapes. The shapes that are studied under mensuration are - circle, rectangle, polygon, triangle, square, cube, cuboid, pyramid, cylinder, cone, sphere, hemisphere etc.

Here, we will study about cones in detail. A cone is a three-dimensional figure having a circular base and tapering to one single point. This point is known as the **vertex **of the cone. The **axis **of the cone is a straight line which joins the vertex of the cone and the center of the circular base.

Above diagram demonstrates a cone in which the radius of the base is shown by "**r**". Its height is referred as the axis of the cone and is denoted by "**h**". The length of lateral surface of the cone is shown by letter "**l**".

Volume of a cone is the measurement of the occupied units of a cone. In other words, the volume of a cone is the estimation of space occupied inside it. The volume of a Cone is represented by cubic units like cubic meter, cubic centimeter, cubic millimeter and so on. Volume of a cone is the number of units used to fill it.

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The volume of a cone formula is given as below:

**Volume of a right circular cone = `(pir^2h)/(3)` **

Where, r = Radius of the circular base

h = Height of the cone

and value of $\pi$ = 3.14

Using this formula, we can find the volume of any cone whose radius and height is given.

In elementary geometry, cone is used as a right circular object.

Volume of the cube is number of cubic units that will completely fill a cone.Volume of a circular cone = $\frac{1}{3}$ (Base area $\times$ Height)

Where, Base Area = $\pi$ r$^2$

Volume of a right cone = $\frac{1}{3}$ $\pi$ r$^2$h

The volume of a cone is one-third the volume of a cylinder having the same base and equal height. The volume of the frustum of the cone is $\frac{1}{4}$ $\pi$ times the frustum of the pyramid.

If r and R are the radii of ends, h is height of a cone, the volume of frustum of a cone is given below:

Volume = $\frac{1}{3}$ $\pi$ h[Rr + R$^2$ + r$^2$] cubic units

Where r and R are the radii of ends and h is height of frustum of a cone.

Volume of cone (V) = $\pi$ $\int_0^h$ y dx

= $\pi$ $\int_0^h$($\frac{-r}{h}$ x + r)$^2$ dx

= $\pi$ $\int_0^h$($\frac{r^2}{h^2}.x^2 + r^2 - \frac{2r^2}{h}. x)^2$ dx

= $\pi$ [$\frac{r^2}{h^2} \times \frac{x^3}{3} + r^2 x - \frac{r^2}{h} \times x^2]_0^h$

= $\pi$ [$\frac{r^2}{h^2} \times \frac{h^3}{3} - \frac{r^2}{h} \times h^2 + r^2 h]$

= $\frac{1}{3}$ $\pi r^2$ h

Hence Proved.

Below are steps for finding the volume of a cone: If radius and height of the cone is given.

Step 1: Square the radius and multiply it with height.

Step 2: Multiply the result of step 1 with $\frac{1}{3}$ $\pi$ (Use $\pi$ = 3.14).

Step 3: Write your answer in cubic units.

Given below are some solved examples on volume of a cone

The volume formula of cone is,

V = `(pir^2h)/3`

V = `((3.14) * (6^2) * (8))/3`

= `((3.14) * (36) * (8))/3`

= `((113.04) * (8))/3`

= `904.32 / 3`

= 301.44

= 301.44

The volume of the cone = 301.44 cm^{3}.

The volume of a cone formula is,

V = `(pir^2h)/3`

V = `((3.14) * (8^2) * (5))/3`

= `((3.14)*(64)*(5))/3`

= `((200.96)*(5))/3`

= `(1004.8)/3`

= 334.93

= 334.93

Volume of the cone = 334.93 cm^{3}.

Radius of base = $\frac{1}{2}$ $\times$ 20 = 10

=> Radius of base (r) = 10 m

Volume of cone = $\frac{1}{3}$ $\pi$ r$^2$ h

200 = $\frac{1}{3}$ $\pi$ (10)$^2$ h

200 = $\frac{1}{3}$ $\pi$ $\times$ 100 $\times$ h

2 = 1.047 $\times$ h

h = $\frac{2}{1.047}$ = 1.91

Hence the height of the cone is 1.91 m.

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