# CURVED SURFACE AREA AND TOTAL SURFACE AREA OF CONE

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.

## Formula for Curved Surface Area and Total Surface Area of Cone

 Curved surface area  =  ΠrlTotal surface area of  =  Πr(l + r)

## Curved Surface Area and Total Surface Area of Cone - Examples

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 cm2

height  =  √(l2 - r2)

=  √(202 - 162)

=  √144

h  =  12

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

Curved surface area =  Πrl

=  Π ⋅ 12 ⋅ 20

=  240Π cm2

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 cm2 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 cm2

Area for 4 persons  =  22(4)  =  88

Πr2  =  88

(22/7) ⋅ r2  =  88

r2  =  88 ⋅ (7/22)

r  =  2√7

height  =  √(l2 - r2)

=  √(192 - (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 cm2, 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 =  √(r2 + h2)

=  √(52 + 122)

=  √(25 + 144)

h  =  √169

h = 13

Area of sheet  =  5720 cm2

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, r =  3

Height of 1st cone  =  3(radius of smaller cone)

=  3(1)  =  3

Height of 2nd cone  =  3(radius of smaller cone)

=  3(1)  =  3

Slant height of 1st cone  =   √(r2 + h2)

=   √(12 + 32)

=   √10

Slant height of 1st cone  =   √(r2 + h2)

=   √(32 + 32)

=   √18  =  3√2

Curved surface area  =  Πrl

Πr1l1 Πr1l1

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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