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 = π r l**

"r" and "l" stands for radius and slant height 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

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 x 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) **x **2Π R

Here R represents radius of the sector

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

So ,circumference of the base of the cone = 66

2 Π r = 66 ==> 2 **x **(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) **x **2Π R

Here R represents radius of the sector

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

So ,circumference of the base of the cone = 44

2 Π r = 44 ==> 2 **x **(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 cm**².

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