Volume of a Cone first created 04/03/07 - last modified xx/xx/xx Page Author: Ty Harness
Figure 1 - Slicing up a right cone
It's well known from school days that the volume of a cone is:
$V = 1/3 A H$
A is the area of the base circle and H is the apex height as shown in Figure 1.
The trouble with school mathematics is that it's just a lesson in memorization.
Greek mathematics from the likes Archimedes and much later the
calculus of Newton and Leibniz show how to formulate an equation for
volume[1].
Accepting the fact the area of a circle is $pi r^2$ which can also be worked out
with calculus and with reference to figure 2 then the volume of thin disc is:
Figure 2 - Taking a thin slither from a cone
$dV = pi r^2 dh $
From figure 2 you can imagine a cone made up of circular discs with a
thickness dh and the
principles of calculus allow you
to accept dh as an infinitesimal slither and by adding
up all the discs
$V = int_0 ^H dV = int_0 ^H pi r^2 dh $
The radius r can be described by a straight line with a gradient of R/H:
note that $pi R^2$ is the Area what some people call the base circle.
From figure 1 you can imagine the circular discs been infinitesimally small then
sliding the cone discs apart (they remain circular) you can create what they call an
oblique cone and hence it will still have the same volume of $ (pi R^2 H) / 3 $
