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.


curved-surface-cone

\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:

  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 $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