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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