# 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 out, 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 $2\pi r$.

Where
$l =$ Slant height of the cone
$r =$ Radius of the base of the cone

Total Surface Area:

Total surface area $=$Area of curved surface $+$Area of base
$\therefore$     $S = \pi rl + \pi {r^2} = \pi r\left( {l + r} \right)$
Rule:

1. The curved surface area of a right circular cone equals the perimeter of the base times one-half slant height.
2. 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 dollar 2 per cubic meter and also the cost of whitewashing the curved surface at $50$cent 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 Meter
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 Meter
Cost of white washing                      $= 277.2 \times \frac{{50}}{{100}} = 138.60$ Dollar