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

AC^{2} = AO^{2} + OC^{2}

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

l = √(r^{2} + h^{2})

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

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)

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

The volume of Right Circular Cone = 1/3 𝜋 r^{2} 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.

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

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

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

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

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.

FAQ (Frequently Asked Questions)

1. What is the Frustum of a Cone?

Answer: Take a right circular cone cut into two such that the cut is parallel to the base, we get two parts the upper part with vertex and the lower part with base. The lower part with the base is called the frustum of the cone. (image will be updated soon)

The Volume of Frustum of Cone =1/3 𝜋 h [(R^{3} - r^{3})/r]

where R is the radius of the lower base, r is the radius of the upper base, and h is the height.

2. What is the Difference Between a Cone and a Right Circular Cone?

Answer: A cone is the simple 3-dimensional figure. A cone has one circular base and one curved surface.

Whereas a Right Circular Cone is the cone in which the line joining the vertex and the center of the base makes a perpendicular at the radius of the base. A Right Circular can be a cone but a cone cannot always be a Right Circular Cone.