Cone is an three-dimensional geometric shape that formed by the locus of all straight line segments to a point called the apex or vertex. The volume of a cone is one-third the volume of a cylinder having the same base and equal height and also the volume of a sphere is equal to the volume of a cone with base of area four times the area of a great circle of the sphere, and with height equal to the radius of the sphere.

The volume of a cone is one third of the product of the area of base and the height of the cone. Height is the perpendicular distance from the base to the apex. The volume of the cone depend upon the area of base and height of the cone.

Cone

Formula:

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

Where,
r = Radius of the cone
h = Height of the cone.

The amount of space occupied by the object is called volume. The capacity of the object that how much it hold by finding its volume. It unit of measurement is cubic units. The volume of a cone is one third of the product of the area of base and the height of the cone.

Steps for finding the Volume of a Cone:

Step 1:
Square the radius and multiply it with $\pi$.

Step 2:
Multiply the result of step 1 with height of the cone.

Step 3:
Multiply the answer in step 2 by $\frac{1}{3}$.


Example 1:

Find the volume of a cone with height 6 and diameter 7.

Step 1:

Height of a cone = 6
Diameter of a cone = 6
Therefore radius of a cone = 3
(The radius of the cone is half the diameter)

Step 2:


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

= $\frac{1}{3} * \frac{22}{7}$$ * 3^2$ * 6

= $\frac{1}{3} * \frac{22}{7}$ * 9 * 6

= 56.57

Hence the volume of a cone is 56.57 cubic units.

Example 2:

If the diameter of the base of a circular cone is 12 cm and its slant height is 10 cm. Find the volume of cone.

Solution:
Step 1:

Diameter of the cone = 12 cm
Therefore radius of cone (r) = 6 cm
Slant height (l) = 10 cm

Step 2:

Find the height of the cone:

we know that,

Slant height (l) = $\sqrt{r^2 + h^2}$

=> 10 = $\sqrt{6^2 + h^2}$

=> 10 = $\sqrt{36 + h^2}$

Solve for h,
By squaring both sides, we have

=> 102 = ($\sqrt{36 + h^2}$)2

=> 100 = 36 + h2

=> 100 - 36 = h2

=> 64 = h2

or h = 8

Step 3:


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

= $\frac{1}{3} * \frac{22}{7}$$ * 6^2$ * 8

= $\frac{1}{3} * \frac{22}{7}$ * 36 * 8

= 301.71

Hence the volume of a cone is 301.71 cm3 .
The volume of the frustum of a cone is equivalent to the sum of the volume of three cones, having for their common altitude and for their several bases, the bases of the frustum and a mean proportional between the bases. The volume of the frustum of the cone is $\frac{1}{4} $$\pi$ times the frustum of the pyramid.

Frustum of a Cone

Formula:

Volume of the frustum of a cone = $\frac{1}{3}$$\pi$h($r^2 + rR + R^2$)

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


Example:
Find the volume of a frustum of a cone with height 5 cm, and radii 3 cm, 7 cm.

Step 1:

Height of frustum of a cone (h) = 5 cm
Radius of top part (r) = 3 cm
Radius of base (R) = 7 cm

Step 2:


Volume of the frustum of a cone = $\frac{1}{3}$$\pi$h($r^2 + rR + R^2$)

= $\frac{1}{3}$$\pi$ * 5($3^2 + 3 * 7 + 7^2$)

= $\frac{1}{3}$$\pi$ * 5(9 + 21 + 49)

= $\frac{1}{3}$$\pi$ * 5(79)

= $\frac{1}{3}$$\pi$ * 395

= 131.7 $\pi$.

Hence the volume of the frustum of a cone is 131.7$\pi$ cm3.