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Volume of a Cone

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.
Cone
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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Volume of a Cone Formula

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Volume of a Cone Formula

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 a Right Cone

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Volume of the cube is number of cubic units that will completely fill a cone. If 'r' is the radius of the base and 'h' is the height (the distance between the apex and the center of the base) of the cone, then
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

Volume of a Truncated Cone

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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.

Truncated Cone Volume

Volume of Frustum of Cone


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.

Derive Volume of a Cone

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Proof of Volume of a Cone

Volume of a Cone Proof


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.

Find the Volume of a Cone

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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.

Volume of a Cone Example

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Given below are some solved examples on volume of a cone

Example 1: Find the volume of cone, whose radius is 6 cm and height is 8 cm.
Solution: 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
The volume of the cone = 301.44 cm3.

Example 2: Find the volume of cone, whose radius is 8 cm and height, is 5 cm.
Solution:
Volume of a Cone Examples
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

Volume of the cone = 334.93 cm3.

Example 3: The volume of solid cone is 200 m$^3$ and diameter of its base is 20 m. Find the height of the cone.

Solution: Let the height of the cone be h m.

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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