**Curved Surface Area and Total Surface Area of Cone :**

Here we are going to see, some example problems of finding curved surface area and total surface area of cone.

Curved surface area = Πrl Total surface area of = Πr(l + r) |

**Question 1 :**

A right angled triangle PQR where angle Q = 90 degree is rotated about QR and PQ. If QR = 16 cm and PR = 20 cm, compare the curved surface areas of the right circular cones so formed by the triangle.

**Solution :**

radius of cone = 16 cm

Slant height = 20 cm

If the triangle is rotated about PQ :

Curved surface area = Πrl

= Π ⋅ 16 ⋅ 20

= 320Π cm^{2}

height = √(l^{2} - r^{2})

= √(20^{2} - 16^{2})

= √144

h = 12

If the triangle is revolved about QR, then radius will be 12 cm

Curved surface area = Πrl

= Π ⋅ 12 ⋅ 20

= 240Π cm^{2}

Hence curved surface area of cone is larger when it is revolved about PQ.

**Question 2 :**

4 persons live in a conical tent whose slant height is 19 cm. If each person require 22 cm^{2} of the floor area, then find the height of the tent.

**Solution :**

Slant height of conical tent = 19 cm

Required area of floor for 1 person = 22 cm^{2}

Area for 4 persons = 22(4) = 88

Πr^{2} = 88

(22/7) ⋅ r^{2} = 88

r^{2} = 88 ⋅ (7/22)

r = 2√7

height = √(l^{2} - r^{2})

= √(19^{2} - (2√7)^{2}

= √[361 - 4(7)]

h = √333

h = 18.25 cm

**Question 3 :**

A girl wishes to prepare birthday caps in the form of right circular cones for her birthday party, using a sheet of paper whose area is 5720 cm^{2}, how many caps can be made with radius 5 cm and height 12 cm.

**Solution :**

r = 5 cm, height of cap = 12 cm

slant height = √(r^{2} + h^{2})

= √(5^{2} + 12^{2})

= √(25 + 144)

h = √169

h = 13

Area of sheet = 5720 cm^{2}

Curved surface area of one cap = Πrl

= (22/7) ⋅ 5 ⋅ 13 -----(1)

Required number of cups = 5720 / [ (22/7) ⋅ 5 ⋅ 13]

= (5720 ⋅ 7)/(22 ⋅ 5 ⋅ 13)

= 28 cups

**Question 4 :**

The ratio of the radii of two right circular cones of same height is 1:3. Find the ratio of their curved surface area when the height of each cone is 3 times the radius of the smaller cone.

**Solution :**

r_{1 } = 1, r_{2 } = 3

Height of 1^{st} cone = 3(radius of smaller cone)

= 3(1) = 3

Height of 2^{nd} cone = 3(radius of smaller cone)

= 3(1) = 3

Slant height of 1st cone = √(r^{2} + h^{2})

= √(1^{2} + 3^{2})

= √10

Slant height of 1st cone = √(r^{2} + h^{2})

= √(3^{2} + 3^{2})

= √18 = 3√2

Curved surface area = Πrl

Πr_{1}l_{1 }: Πr_{1}l_{1}

1√10 : 3 (3√2)

√10 : 9√2

√5 : 9

After having gone through the stuff given above, we hope that the students would have understood, "Curved Surface Area and Total Surface Area of Cone".

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