## About "Curved surface area of cone"

Curved surface area of cone is the measurement of outer area,where the extension of bottom portion wont be included.

## Curved surface area of cone

Cone is a solid or hollow object which tapers from a circular or roughly circular base to a point.

CSA of cone = π r l

"r" and "l" stands for radius and slant height of cone.

## Example problems of curved surface area of cone

Question 1 :

If the vertical angle and the radius of the right circular cone are 60 degree and 15 cm respectively,then find its slant height and curved surface area.

Solution :

Vertical angle of the  right circular cone = 60°

radius of the cone (r) = 15 cm

In the triangle ABC, ∠ABC = 30°

BC = 15 cm

Opposite side (BC) = 15 cm

Hypotenuse side (AC) = ?

here,  we need to find the measurement of the side AC. So we have to use Sinθ.

Sin θ = Opposite side/Hypotenuse side

Sin 30° = BC/AC

(1/2) = 15/AC

AC = 30 cm

Slant height (L) = 30 cm

Curved surface area of cylinder  =  π r l

=  π  15 x 30 ==> 450 π cm²

Hence, Curved surface area of cone is 450 π cm²

Question 2 :

If the circumference of the base of the solid right circular cone is 236 and its slant height is 12 cm, find its curved surface area.

Solution :

Circumference of the base = 236 cm

Slant height (L) = 12 cm

2 Π r = 236 ==> Π r = 236/2 ==> Π r = 118

Curved surface area of cone = Π r l

=  118 (12) ==> 1416 cm²

Hence, curved surface area of cone is 1416 cm²

Question 3 :

A heap of paddy is in the form of a cone whose diameter is 4.2 m and height is 2.8 m. If the heap is to be covered exactly by a canvas to protect it from rain,then find the area of the canvas needed.

Solution :

Diameter of heap of paddy = 4.2 m

r = 4.2/2 =  2.1 m

height of paddy (h) = 2.8 m

L² = r²  + h² ==> L = √(2.1)² + (2.8)²  ==> L = √4.41 + 7.84

L = √12.25 ==> L = √ 3.5 x 3.5 ==> L = 3.5 cm

Curved surface area of heap of paddy = Π r l

= (22/7) x (2.1) x (3.5) ==> 22 x (2.1) x (0.5) ==> 23.1 cm²

Hence, curved surface area of paddy = 23.1 cm²

Question 4 :

The central angle and radius of a sector of a circular disc are 180 degree and 21 cm respectively. If the edges of the sector are joined together to make a hollow cone,then find the radius of the cone.

Solution:

The cone is being created by joining the radius. So the radius of the sector is going to be the slant height of the cone.

Slant height (L) = 21 cm

Arc length of the sector = Circumference of the base of the cone

Length of arc = (θ/360) 2Π R

Here R represents radius of the sector

=  (180/360) 2 x (22/7)(21) ==> (1/2) 2 x 22 3 ==>  66 cm

So ,circumference of the base of the cone = 66

2 Π r = 66 ==> (22/7) x r = 66 ==> 10.5 cm

Hence, radius of the cone = 10.5 cm

Question 5 :

Radius and slant height of a solid right circular cone are in the ratio 3:5. If the curved surface area is 60 Π cm², then find its radius and slant height.

Solution :

Radius and slant height of a solid right circular cone are in the ratio 3:5.

r : L = 3 : 5 ==>  r / L = 3 / 5 ==> r = 3L / 5

Curved surface area of cone = 60 Π cm²

Π r L = 60 Π ==> Π x (3L/5) x L = 60 Π ==> L² = 60 Π x (1/Π) x (5/3)

L² = 60 x (5/3) ==>  L² = 100 ==> L = 10 cm

r = 3(10)/5 ==> 30/5 ==> 6 cm

Hence, radius and slant height of cone are 6 cm and 10 cm respectively.

Question 6 :

A sector containing an angle of 120 degree is cut off from a circle of radius 21 cm and folded into a cone. Find the curved surface area of a cone.

Solution :

The cone is being created by joining the radius. So the radius of the sector is going to be the slant height of the cone.

Slant height (L) = 21 cm

Arc length of the sector = Circumference of the base of the cone

Length of arc = (θ/360) 2Π R

Here R represents radius of the sector

=  (120/360) 2 x (22/7)(21) ==> (1/3) 2 x 22 3 ==>  44 cm

So ,circumference of the base of the cone = 44

2 Π r = 44 ==> 2 (22/7) x r = 44 ==> 7 cm

Now, we need to find the curved surface area of cone

CSA of cone = Π r l ==> (22/7) x 7 x 21 ==> 462 cm²

Hence, CSA of cone is 462 cm².

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