Integral Calculus 201-NYB-05
Vincent Carrier
Archimedes and the Area of the Circle
A regular polygon is a polygon whose sides all have equal length. Let us consider a
sequence of regular polygons with n sides inscribed in a circle.
n=3
n=4
n=5
n=6
n=7
n=8
n=9
n = 10
n = 15
n = 20
As is clear from the drawings, the higher the value of n, the closer the approximation of
the circle by the polygon. Indeed, for n as small as 20, the polygon and the circle are
almost indistinguishable.
Let
A : area of the circle
An : area of the regular n-sided polygon inscribed in the circle.
Then
lim An = A.
n→∞
Consider a regular n-sided polygon inscribed in a circle of radius r.
r
r sin
π/n
r cos
π n
π n
The area of the triangle on the right is
π i h
π i
π π 1h
r2
r sin
r cos
=
sin
cos
.
2
n
n
2
n
n
There are 2n such triangles in the polygon. Therefore,
2
π π π π r
sin
cos
= nr2 sin
cos
An = 2n
2
n
n
n
n
and
π i
h
π lim nr2 sin
cos
n→∞
n
n
h
π i h
π i
= r2 lim n sin
lim cos
n→∞
n→∞
n
n
π i
sin(π/n) h
2
lim cos
= r π lim
n→∞
n→∞
π/n
n
lim An =
n→∞
= πr2 .
The idea of the above proof, i.e. approximating a circle by a sequence of polygons, can
be applied fruitfully with slight modifications to the problem of finding the area of any
curved surface.
The formula above was first found by the Greek mathematician and scientist Archimedes
of Syracuse (287-212 BC) using a similar method. He was also the first to find the formula
for the volume (V ) and the surface area (A) of a sphere of radius r:
4
V = πr3
3
A = 4πr2 .
According to legend, he had a sphere inscribed in a cylinder put on his tombstone.
Sphere
r
It can be seen that
Cylinder
4πr3
3
VC = 2πr3
AS = 4πr2
AC = 6πr3
VS =
VC
AC
3
=
= .
VS
AS
2