Surface Area of a Cone

Surface Area of a Cone

Surface Area of a Cone

Surface area of a cone can be measured on the basis of its slant height and the radius of the base. There are two types of surfaces on a solid cone, one is the curved surface around the cone and another is the plane surface of the circular base.
Curved surface and plane surface of cone
Curved surface area of a cone is given by the formula πrl, where r is the radius and l is the slant height of the cone. This formula can be understood and derived by the following activity:
Activity to derive the formula of curved surface area of a cone
1.   Take a hollow right circular cone of vertical height h, radius r and slant height l.
2.   Cut this cone along the slant height and spread it out. The spread out figure is a sector of circle of radius l (slant height of the cone). The arc of this sector is equal to the circumference of the base of the cone.
3.   Divide the sector in many small parts as you can and arrange as shown in the figure.

In this way we can get a rectangle of length l and breadth πr as shown in the figure.
Now,
Curved surface area of the cone = Area of the sector given in the figure
                                                    = Area of rectangle
                                                    = length × breadth
                                                    = l × πr
                                                    = πrl

   Curved surface area of the cone = πrl

And,
Total surface area of the cone = Curved surface area of cone + Area of circular base
                                                = πrl + πr2
                                                = πr (r + l)

   Total surface area of the cone = πr (r + l).

Workout Examples

Example 1: Find the total surface area of a cone whose radius = 5 cm and height = 12 cm.
Example 1 Cone
Solution: Here,
                                Radius of cone (r) = 5 cm
                                Height (h) = 12 cm                               
Slant height of the cone

                We know,
                                Total surface area of the cone = πr(r+l)
                                                                                    = 22/7 × 5 (5 + 13)
                                                                                    = 22/7 × 5 × 18
                                                                                    = 282.85 cm2


Example 2: The circumference of the base of tent is 17.6 m. The slant height is 3.5 m. Find the area of the canvas used for tent.

Solution: Here,
                           Circumference (c) = 17.6 m
                i.e.      2πr = 17.6
                or,      πr = 17.6/2
                or,      πr = 8.8 m

                          Slant height (l) = 3.5 m

                Area of canvas required = Curved surface area of the cone
                                                          = πrl
                                                          = 8.8 × 3.5
                                                          = 30.8 m2


Example 3: A cone with hemisphere on the top has common radius 21 cm. If the slant height of the cone is 75 cm, find the total surface area of the solid object.
Combined solid of cone and hemisphere
Solution: Here,
                           Common radius (r) = 21 cm
                           Slant height of the cone (l) = 75 cm

                Now,
                           Total suface area of the solid (A) = CSA of hemisphere + CSA of cone
                                                                                   = 2πr2 + πrl
                                                                                   = πr (2r + l)
                                                                                   = 22/7 × 21 (2×21 + 75)
                                                                                   = 7722 cm2

                Thus, the TSA of the given solid is 7722 cm2.


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