Right Circular Cone

In our day to day life, we come across shapes like ice cream cones, a clown’s cap, a tent, etc, these all have the right circular cone shape.

Right Circular Cone can be defined as, 

A right circular cone is a solid generated by revolving a line segment that passes through a fixed point and which makes a constant angle with a fixed-line.(image will be updated soon)

Here you see a right circular cone with ‘h’ is the height of the cone, ‘r’ is the radius of the cone, and L is the slant height of the cone.

Some important terms related to the right circular cone formula. And let us see the different equations of the right circular cone.(image will be updated soon)

• Vertex: The fixed point A is called the vertex of the cone.

• Axis: The fixed-line AO is called the axis of a cone.

• Base: A right circular cone has a circular plane end. This is called the base of a cone. Where O is the center. And OC is the radius of the base.

• Height: The length of the line segment joining the vertex and the center of the base is called the height of the cone. It is denoted by ‘h’

• Slant Height: The length of the line segment joining the vertex to any point on the edge of the base, is called the slant height of the cone. It is denoted by ‘l’

• Radius: The length of the line segment joining the center and any point on the edge of the base is the radius of the base.it is denoted by ‘r’.

Let us see some formulas used to solve problems on Right Circular Cone.

Slant Height of the Right Circular Cone (image will be updated soon)

Consider a right circular cone of height AO = h, slant height AC = l, and radius OC = r. Angle AOC = 900.

By Pythagoras theorem, we have

AC² = AO² + OC²

l² = h² + r²

l = √(r² + h²)

It follows from this relation of height, slant height, and radius we can find any of them.

Surface Area of Right Circular Cone

The curved surface area of a cone is also called as the lateral surface area.

The curved surface area of a cone  S with radius r, height h, and slant height h is

S= π r l 

S = ½ ( 2πr) l

S = ½ ( circumference of base ) x (slant height)

Now, Total Surface area of right circular cone =  Curved surface area + Area of the base

= S + πr2
= πrl + πr2

= πr( l + r)

Volume of Right Circular Cone

To find the volume of Right Circular Cone with radius r and height h 

The volume of Right Circular Cone = 1/3 𝜋 r² h

= 1/3 𝜋 r2 h

= 1/3 x ( area of the base) x height

Now let us see some problems using the right circular cone formula. Memorize the equation of the right circular cone and solve the problems on the right circular cone easily.

Solved Examples

Example 1:The radius of a cone is 3 cm and the vertical height is 4cm. Find the area of the curved surface of the right circular cone.

Solution: We have r = 3cm, h = 4 cm

Let h cm be the slant height of the cone.

Then Slant height  \[ = \;{r^2} + {h^2}\]

\[ = {3^2} + {4^2}\]

= 9 + 16

= 25

l = 5cm

Area of the curved surface =πrl

\[ = {\text{ }}\frac{{22}}{7} \times 3 \times 5\]

= 47.14 cm2

Example 2: The diameter of a right circular cone is 8 cm and its volume is \[48\pi c{m^3}\] What is its height?

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

The diameter of the cone = 8 cm

The radius of the cone = 4 cm

Volume of the cone = \[48\pi c{m^3}\]

Now, 

\[\frac{1}{3} \times \pi  \times 4 \times 4 \times h = 48\pi \]

\[h = \frac{{48\pi  \times 3}}{{16\pi cm}}\]= 9cm

Hence the height of the cone is 9 cm. 

Example 3: A solid cube of side 7 cm is melted to make a cone of height 5 cm, find the radius of the base of the cone.

Solution: We have, 

Volume of the cube = \[{\left( {Side} \right)^3} = {7^3}c{m^3} = 343{\text{ }}c{m^3}\]

Let the radius of the base be r cm. Then, 

Volume of the cone \[ = {\text{ }}\frac{1}{3} \times \frac{{22}}{7} \times {r^2} \times 5{\text{ }}c{m^3}\]

Since the solid cube is melted to make a cone    

Volume of the cube = Volume of the cone

343 = ⅓ x 22/7 x r2 x 5

r2 =  343 x 3 x 7 / 22 x 5

r2  = 7203/110 = 65.48

r = 8.09

Quiz Time

1. The radius of the cone is 5cm and the vertical height is 12cm. Find the area of the curved surface.

2. Find the curved surface area of a cone with base radius 5cm and the slant height 7cm.

3. Find the total surface area of a right circular cone, its slant height is 21cm, and the diameter of its base is 24cm.

4. A conical vessel whose internal radius is 5cm and height is 24cm is full of water. The water is poured into a cylindrical vessel with an internal radius of 7cm. Find the height to which the water rises.