Cone
A cone is a pyramid with circular base. Take a circular
piece of paper with centre O. cut off the sector APB and join the edges AO and
BO. In this way a pyramid with a circular base is formed which is called a
circular pyramid or a cone.
A cone has a curved surface and a circular base. In the
figure given below, O is the centre of the circular base, OA is the radius (r),
OP is the vertical height (h) of the cone and PA is the slant height (l).
In right angled triangle AOP, using Pythagoras Theorem,
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Surface Area and Volume of Cone
A cone is formed from the sector of a circle. So its
curved surface area is the surface of the sector.
1. The
curved surface area of a cone = Area of the sector = πrl, where r is the radius of the circular base
and l is the slant height.
2. The
total surface area of a cone = curved surface area + area of circular base
= πrl + πr2
= πr
(r + l)
3. Volume
of the cone = volume of the circular pyramid
= 1/3 of area of circular base ×
height
=
1/3 × πr2 × h
= πr2h/3
Workout Examples
Example 1: Calculate the curved sirface area, total surface area and
volume of the given cone.
Solution: Here,
Height
of cone (h) = 8 cm
Slant
height (l) = 10 cm
Now,
Curved
surface area (CSA) = πrl
= 22/7 × 6 × 10
= 188.57 cm2
Total
surface area (TSA) = πr(r + l)
= 22/7 × 6 (6 + 10)
=
301.71 cm2
Volume
of cone (V) = πr2h/3
= 1/3 × 22/7 × 62 × 8
= 22/21 × 36 × 8
= 301.71 cm3
Example 2: If the total surface area of a cone is 704 cm2
and radius of its base is 7 cm, find the volume of the cone.
Solution: Here,
Radius
of cone (r) = 7 cm
Total
surface area of cone = 704 cm2
i.e. πr (r + l) = 704
or, 22/7 × 7 (7 + l) = 704
or, 22 (7 + l) = 704
or, 154 + 22l = 704
or, 22l = 704 – 154
or, l = 550/22
or, l = 25 cm
Now,
Volume
of cone (V) = πr2h/3
= 1/3 × 22/7 × 72 × 24
= 1232 cm3
Example 3: If the volume of the given cone is 1848 cm3,
and the radius of its base is 14 cm, find its curved surface area.
Solution: Here,
Radius
of cone (r) = 14 cm
Volume
of cone = 1848 cm3
i.e. πr2h/3 = 1848
or, 1/3 × 22/7 × 142 × h = 1848
or, 22/21 × 196 × h = 1848
or, h = 1848 × 21/4312
or, h = 9 cm
Now,
Curved
surface area of cone (CSA) = πrl
= 22/7 × 14 × 16.64
= 732.16 cm2
You can comment your questions or problems
regarding the surface area and volume of cone here.
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