Curved Surface Area of a Cone
If a perpendicular cut is made from a point on the circumference of the base to the vertex and the cone is opened up, a sector of a circle with radius $$l$$ is produced. Since the circumference of the base of the cone is $$2\pi r$$, therefore the arc length of the sector of the circle is $$2\pi r$$.
\[\begin{gathered} {\text{Curved surface area}} = {\text{Area of sector }}OAA’ \\ {\text{Curved surface area}} = \left( {\frac{{{\text{Arc length of sector}}}}{{{\text{Circumference of circle}}}}} \right) \times {\text{Area of circle}} \\ {\text{Curved surface area}} = \frac{{2\pi r}}{{2\pi l}} \times \pi {l^2} = \pi rl \\ \end{gathered} \]
Where
$$l = $$ slant height of the cone
$$r = $$ radius of the base of the cone
Total Surface Area
The total surface area $$ = $$area of curved surface $$ + $$area of base
$$\therefore $$ $$S = \pi rl + \pi {r^2} = \pi r\left( {l + r} \right)$$
Rule:

The curved surface area of a right circular cone equals the perimeter of the base times onehalf slant height.

The total surface area equals the curved surface area of the base.
Example:
The slant height of a conical tomb is $$10.5$$ m. If its diameter is $$16.8$$ m, find the cost of cleaning it at $2 per cubic meter and also the cost of whitewashing the curved surface at $$50$$ cents per square meter.
Solution:
Now, slant height, $$l = 10.5$$ m
Perpendicular height $$h = \sqrt {{{\left( {10.5} \right)}^2} – {{\left( {8.4} \right)}^2}} = 6.3$$ m (as $$h = \sqrt {{l^2} – {r^2}} $$)
Volume of the conical tomb $$ = \frac{1}{3}\pi {r^2}h$$
$$ = \frac{1}{3} \times \frac{{22}}{7} \times 8.4 \times 8.4 \times 6.3$$
$$ = 465.7$$ cubic meters
Cost of construction $$ = 465.7 \times 2 = 931.39$$
Curved surface $$ = \pi rl = \frac{{22}}{7} \times 8.4 \times 10.6 = 277.2$$ square meters
Cost of white washing $$ = 277.2 \times \frac{{50}}{{100}} = 138.60$$ dollars
charity joseph
July 23 @ 12:12 pm
find in terms of pi, the curved surface area of a cone with circular base diameter 10cm and height 12cm
Prabhat
February 4 @ 5:23 pm
If perpendicular is given and curved surface area of a cone is given, then what’s the formula for finding length?