Curved surface area of cone is the measurement of outer area,where the extension of bottom portion wont be included.
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
Adjacent side (AB) = ?
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 cm^{2}
So, curved surface area of cone is 450 π cm^{2}
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 cm^{2}
So, curved surface area of cone is 1416 cm^{2}
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
l^{2} = r^{2}+h^{2}
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 cm^{2}
So, curved surface area of paddy is 23.1 cm^{2}.
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
2 ⋅ (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
l^{2} = 60 Π ⋅ (1/Π) ⋅ (5/3)
l^{2} = 60 ⋅ (5/3)
l^{2} = 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
2 ⋅ (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 cm^{2}
So, curved surface area is 462 cm^{2}.
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