VOLUME OF RIGHT CIRCULAR CONE – MEANING & FORMULA

There are various things which you have seen in your life such as a volcano, a party hat, an ice-cream cone etc. All these are examples of cones. A cone is a 3-D figure which has a radius and height. In this article, we will be discussing the volume of the right circular cone, its formula and derivation.

VOLUME OF RIGHT CIRCULAR CONE

MEANING OF RIGHT CIRCULAR CONE

A cone is a 3-dimensional figure which has two ends, at one end there is a circular plane(base) and the other end is pointed. When the axis of a cone meets the vertex and joins the midpoint of the circular plane(base) perpendicularly then it is known as the right circular cone.

PROPERTIES OF RIGHT CIRCULAR CONE 

1. A right circular cone is formed when a right-angled triangle is rotated about its perpendicular and the perpendicular becomes the axis of the cone.

2. The axis of the right circular cone is perpendicular to its circular base. Due to this, the axis of the cone overlaps its height.
3. When the vertex and any two points of the circular plane(base) are joined, an isosceles triangle is formed.

VOLUME AND FORMULA

The volume of a right circular cone is the maximum amount of space it can contain. In simple words, it is the capacity of the cone. 

Volume of a cone = ⅓ rd volume of a cylinder.

Formula:

Volume =\frac{1}{3} \pi r^{2}h cubic units

where r = radius of the circular base & h = height of the cone.

As we know, the axis of the cone is perpendicular to the base, therefore, by using Pythagoras’s theorem we get,

l^{2}=r^{2}+h^{2}

l=\sqrt{{r}^{2}+h^{2}}

This is the formula for finding the slant height of the cone. If we know any of the two parameters, we can easily find the third parameter using this formula.

DERIVATION OF VOLUME OF CONE

STEP 1
Take a cylindrical vessel and a conical flask of the same radius and height.

STEP 2
Fill the conical flask fully with water and pour this water into the cylindrical vessel. We will observe that the cylindrical vessel is not filled.

STEP 3
On repeating this process we will see that there is again some volume left in the cylinder.

STEP 4
When the same is done the third time we will observe that the cylindrical vessel is filled.

Hence, the volume of a cone = 1⁄3 volume of a  cylinder.

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

where r = radius of the circular base & h = height of the cone.

ILLUSTRATION

Q1. If the height of the cone is 14 cm and the diameter is 18 cm. Find the volume of the cone.

Solution:

radius(r) = Diameter/2

                = 18/2 cm

                = 9 cm

Volume of cone

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

Q2. The slant height of the cone is 13 cm and the diameter is 10 cm. Find the volume of the cone.

Solution:

radius(r) = Diameter/2

                = 10/2 cm

                = 5 cm

We know that

\mathrm{l}^{2} =\mathrm{r}^{2}+\mathrm{h}^{2} ( \mathrm{l}= slant height)

13^{2} =5^{2}+\mathrm{h}^{2}
\Rightarrow \mathrm{~h}^{2} =169-25
\Rightarrow \mathrm{~h} =\sqrt{144}
\therefore \mathrm{~h} =12 \mathrm{~cm}

Volume of cone

=\frac{1}{3}\pi r^{2}  h
=\frac{1}{3}\times 3.14 \times 5^{2}\times 12           (π = 3.14)
=314 \mathrm{~cm}^{3}