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). Based on these values, formulae for the cone’s surface area and volume have been used. A cone is a three-dimensional geometric shape that narrows gradually from a flat base (typically circular base) to a point called the apex or vertex (Which creates an axis to the center of the base).

Formulas Related to Cone 

A cone is a geometric form with a curving surface and a circular base. The following properties of a cone help in its identification. These are mentioned below.

• A cone’s base is circular.
• A cone has one face, one vertex, and no edges.
• A cone’s slant height is the length of the line segment from the apex of the cone to any point on the circle of the cone’s base.
• A right circular cone is one that has its apex right above the circular base at a perpendicular distance.
• An oblique cone is one with an apex that is not directly above the circular base.

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 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 (l) = √(r²+ h²)

• 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, The formula for the curved surface area of a Cone,

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 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 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 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 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 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