## 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 = πrl

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

Example 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  30

=  450 π cm2

So, curved surface area of cone is 450 π cm2

Example 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

Πr  =  236/2

Π r  =  118

Curved surface area of cone  =  Π r l

=  118 (12)

=  1416 cm2

So, 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

r  =  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 3.5)

l  =  3.5 cm

Curved surface area of heap of paddy  =  Π r l

=  (22/7) ⋅ 2.1 ⋅ 3.5

=  22 ⋅ 2.1 ⋅ 0.5

=  23.1 cm2

So, curved surface area of paddy is 23.1 cm2.

Example 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 ⋅ (22/7) ⋅ 21

=  (1/2)  2 ⋅ 22 ⋅ 3

=  66 cm

So, circumference of the base of the cone = 66

2 Π r  =  66

⋅ (22/7) ⋅ r = 66

r  =  10.5 cm

So, radius of the cone is 10.5 cm.

Example 5 :

Radius and slant height of a solid right circular cone are in the ratio 3:5. If the curved surface area is 60Π 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 Πcm²

Π r L  =  60Π

Π ⋅ (3L/5) ⋅ L  =  60Π

l2  =  60 Π  (1/Π)  (5/3)

l2  =  60 ⋅ (5/3)

l2  =  100

l  =  10 cm

r  =  3(10)/5

r  =  30/5

r  =  6 cm

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

Example 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 ⋅ (22/7)  21

=  (1/3)  2  22 ⋅ 3

=  44 cm

So, circumference of the base of the cone = 44

2Πr  =  44

(22/7)  r  =  44

r   =  7 cm

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

Curved surface area of cone  =  Πrl

=  (22/7) x 7 x 21

=  462 cm2

So, curved surface area is 462 cm2

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