Surface Area of Cone

In this section we will discuss surface area of cone.

Cone : When we increase the base of square pyramid that base becomes circular and the figure changes to cone .
In real life we use this shape in funnel, joker’s cap etc.
Its base is circular and has a curved surface.



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

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

Base : At the bottom there is a circular shape which is called the base.

Height : The Axis AO is also called the height.

Slant height : The length of the line joining the vertex to any point on the circular edge of the base is called slant height. It is denoted by ‘l’. In the above diagram AC is the slant height.

Radius : O is the center so OC and OB are the radii.

Formulas for surface area of cone and curved surface area of cone :
l2 = r2 + h2
Curved surface area = CSA = π r l
Total surface area = TSA = π r ( r + l )

Some solved examples on surface area of cone

1) The diameter of a cone is 16 cm and its height is 6 cm. Find the surface area of cone.
Solution : Diameter = d = 16 cm ⇒ r = 8 cm ; h = 6 cm
l
2 = r 2 + h 2
l
2 = 8 2 + 6l 2
⇒ = 64 + 36
⇒ = 100
l
2 = 100
∴ l = 10 cm
TSA(surface area of cone) = π r ( r + l )
⇒ = 3.14 x 8 ( 8 + 10)
⇒ = 3.14 x 8 x 18
∴ TSA = 452.16 cm
2
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2) A corn cob somewhat like a cone, has the radius of the broadest end as 2.1 cm and length as 20 cm. If each 1 cm
2 of the surface of the cob carries an average of 4 grains, find how many grains you would find on the entire cob?
Solution :

Total number of grains = CSA of corn cob x Number of grains on 1 cm
2
r = 2.1 cm and h = 20 cm
l = √( r
2 + h 2 )
l = √( 2.1
2 + 20 2 )
⇒ = √( 4.41 + 400)
⇒ = √(404.41)
∴ l = 20.11 cm
CSA = π r l
⇒ = 3.14 x 2.1 x 20.11
∴ CSA of corn cob = 132.73 cm
2
Hence, the total number of grains on the corn cob = 132.73 x 4 = 530.92
So, there would be approximately 531 grains on the corn cob.
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3) A tent is of the shape of a right circular cylinder up to height 3 m and then becomes the right circular cone with a maximum height of 13.5 m above the ground. Calculate the cost of painting the inner side of the tent at the rate of $2 per sq. m., if the radius of the base is 14 m.
Solution :

r = 14 m and h = 3 m
CSA of cylinder = 2 x π x r x h
⇒ = 2 x 3.14 x 14 x 3
∴ CSA of cylinder = 263.76 m
2
For cone :
r = 14 cm and h = 13.5 – 3 = 10.5 m
l = √( r
2 + h 2 )
l = √( 14
2 + 10.5 2 )
⇒ = √( 196 + 110.25 )
⇒ = √(306.25)
∴ l = 17.5 cm
CSA = π r l
⇒ = 3.14 x 14 x 17.5
∴ CSA of cone = 769.3 m
2
∴ Total area which is painted = CSA of cylinder + CSA of cone
⇒ = 263.76 + 769.3 = 1033.06 m
2
Hence, the cost of painting = 1033.06 x 2 = $2066.12 = $ 2066.

Surface Area :

Surface Area of Cube
Surface Area of Rectangular Prism(Cuboid)
Surface Area of Cylinder
Surface Area of Cone
Surface Area of Sphere and Hemisphere
Surface Area of Prism
Surface Area of Pyramid

From Cone to Mensuration

From Cone to Mensuration>