Volume and Surface Area of Cone Formula

A cone is a shape formed by connecting a common point, known as the apex or vertex, to all the points of a circular base using a set of line segments (which does not contain the apex).

• Height (h): The height of the cone is the distance from the vertex to the base.
• Radius (r): The circular base has measured the value of the radius.
• Slant height (l): The slant height is the length of the cone from the apex to any point on the circumference of the base.

Slant height

Volume of Cone 

The volume of the cone (V), where "r" is the radius of its circular base, "h" is the height from the vertex to the base, and "l" is the length of the cone's edge.

Volume(V) of cone  = ⅓ πr²h cubic units

Total Surface Area of Cone 

A right circular cone's surface area is equal to the sum of its lateral surface area (πrl) and circular base surface area (πr²). The formula for the Surface Area of the cone is,

Area = πr(l + r) square units

Curved Surface Area of Cone

A cone's curved surface area is the area enclosed by the curved part of the cone. The curved surface area of a cone with radius 'r', height 'h', and slant height 'l' is as follows:

Area = πrl square units.

Sample Questions

Question 1:  Find the volume of the cone if radius, r = 5 cm, and height, h = 6 cm.

Solution:

Given: Radius = 5 cm 

=Height = 6 cm 

Now we have formula to calculate volume of cone,

V = ⅓ πr²h

V = (⅓) × (22/7) × 5² × 6

V =  (⅓) × (22/7) × 25 × 6 

V = 3300/21

= 157.14 cubic cm 

Therefore the volume of cone is 157.14 cubic cm.

Question 2: What is the total surface area of the cone with a radius of 7 cm and a height of 5 cm?

Solution:

Given: radius = 7 cm and height = 5 cm,

Total surface area of cone is,

Area = πr(l + r)

Since, slant height  l  = √(r² + h²) 

= √(7² + 5²) 

= √(49 + 25) 

= √74

Therefore,

Surface Area of Cone , Area = πr(l + r)

A = π × 7(√74 + 7) 

= π × 7(8.60+ 7) 

= π × 7(15.60) 

= 22/7 × 7(15.60) 

=  343.25 square cm 

Question 3: If the height of a given cone is 5 cm and the diameter of the circular base is 8 cm. What will be the volume of the cone?

Solution:

Diameter of the circular base = 8 cm.

So, radius = 8/2 = 4 cm

Height = 5 cm

By the formula of cone volume, 

Volume of Cone  = 1/3 πr²h

So by putting the above values of r and h in the volume formula 

Volume   = 1/3 π 4² 5

Since, π = 22/7

Therefore the volume of cone is

Volume  = 1/3 × 22/7 × 4² × 5

Volume  = 1/3 × 22/7 × 16 × 5 

= 1760/21

=  83.81 cubic cm 

So the volume of cone is 83.81 cubic cm

Question 4: Find out the slant height if the diameter is 10 cm and the height of the cone is 15 cm.

Solution:

Given: Diameter = 10 cm and height of cone (h) = 15 cm 

To find the slant height (l) = ?

l = √(r² + h²) 

Radius = diameter /2

= 10/2 

= 5 cm 

Therefore, Slant height (l) = √(r² + h²) 

= √(5² + 15²) 

= √(25 + 225)

= √(250)

 = 15.81 cm 

Therefore the slant height of cone is 15.81 cm.

Question 5: If the height of a given cone is 6 cm and the diameter of the circular base is 12 cm. What will be the volume of the cone?

Solution:

Diameter of the circular base = 12 cm.

So, radius = 12/2 = 6 cm

Height = 6 cm

By the formula of cone volume, 

Volume of Cone  = 1/3 πr²h

So by putting the above values of r and h in the volume formula,

Volume = 1/3 π 6² 6

 Since, π = 22/7

Therefore the volume of cone is

Volume = 1/3 × 22/7 × 36 × 6

Volume = 1/3 × 22/7 × 36 × 6 

= 4752/21

= 226.28 cubic cm 

So the volume of cone is 226.28 cubic cm 

Question 6: What will be the curved surface area? If the radius is 2 cm and the height is 5 cm?

Solution:

Given: Radius = 2 cm 

Height = 5 cm 

To find the curved surface area we use formula,

Area = πrl square units.

For this we have to find l, 

l = √(r² + h²) 

= √(2² + 5²) 

= √(4 + 25) 

= √(29) 

= 5.38 cm 

Now, Curved surface area = πrl  

= 3.14 × 2 × 5.38 

= 33.78 square cm 

So the curved surface area of cone is 33.78 square cm