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.

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.