
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 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:
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.
Solution: Here,
Radius
of cone (r) = 5 cm
Height
(h) = 12 cm
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.
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.
You can comment your questions or problems
regarding the surface area of cone here.
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