Answer 1:
As you probably know, the area of a circle is
pi(r^{2})
where pi = 3.14159265... and
r^{2} = r X r where r is the radius (half
the diameter) of the circle.
As you mention, if
you inscribe a regular polygon inside a circle and
circumscribe the circle with a similar regular
polygon with the same number of sides and take the
average of the areas of those two polygons you get
a number that is close to the area of the
circle.
As you would expect, as the number of
sides of the polygons gets large, the average of
the two areas gets closer and closer to the area
of the circle.
Since most people don't
learn the math required to solve this problem
until some time in high school, I'll give a few
examples first.
First, lets say n equals the
number of sides of the polygons used: n =
3 for
triangles, n = 4 for squares, n = 5
for pentagons,
etc.
According to the formula I came up with,
here is the average of the two areas (compare to
pi r^{2}):
n = 3: 3.2476 r^{2}
n = 4: 3 r^{2}
n = 5: 3.0052 r^{2}
n = 6:
3.0311 r^{2}
n = 10:
3.0941 r^{2}
n = 100:
3.1411 r^{2}
n = 1000:
3.14159 r^{2}
As you can see, the
number before r^{2} gets closer and closer
to pi the higher n is. Using just triangles
or squares isn't actually that bad of an
approximation as it turns out.
For those
who have had trigonometry (or have a scientific
calculator to play with), here is the formula I
got for the average of the areas of two regular
polygons with n sides, one that is inscribed in a
circle of radius r, the other circumscribing the
circle: 0.5 n [ tan(pi/n) + sin (pi/n)
cos (pi/n) ] r^{2}
