Cone

Cone

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.
making of 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).
different parts of cone
In right angled triangle AOP, using Pythagoras Theorem,
slant height, vertical height and radius

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 
radius of cone

                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
vertical height of cone

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.
example 3 cone
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               
slant height of cone

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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