Surface Area of a Right Cone first created 27/06/07 - last modified 17/07/07 Page Author: Ty Harness
Figure 1 - Developed Right Cone
The slant height L:
$L = sqrt(R^2 + H^2)$
You can imagine a cone laying on its slant on a table could be rolled around the apex and if you marked
the base circle you could find what sheet metal workers call the developed pattern. Sheet metal workers divide the
base circle into 12,24 or 36 divisions using set squares or dividers and step off (truncate) the same
number of divisions around the base curve.
Mathematically the length of the base curve,s, is the circumference of a circle.
s = $2*pi*R$ which is the radius multiplied the angle subtended ($2*pi$) therefore to find the angle $phi$
$s = L*phi$ or $phi = (2*pi*R)/L$
The area of a small strip:
$delta A = phi*q*dq$
To find the surface area we can turn to calculus and imagine that dq is very very small and summate all
the strips.
$A = int_0 ^L phi*q*dq$
$A = phi*[q^2 /2]_0^L$
$A = phi*L^2 /2 = pi*R*L$
$A = pi*R*sqrt(R^2 + H^2)$
I believe the above is called the lateral surface area in that it doesn't account for the circular base area and you'll
often see in books and on the net that the surface area is quoted as
