This is π, but how did Archimedes calculate
it?
Here is a formula that is derived from Archimedes
method that approximates the value of π.
We can input any positive integer for n, and
the result will give us an approximate value
of π.
The higher the value we input for n, the closer
we get to π.
So now why does this equation work?
Around 200 B.C. Archimedes devised a method
for calculating π by method of exhaustion.
First, we create a circle with a radius of
1/2.
Now remember that the circumference of a circle
is equal to 2πr.
So with a radius of 1/2 the circumference
is equal to π.
Next inside the circle we inscribe a polygon.
And we increase the number of sides repeatedly.
What we can see is that every time we increased
the number of sides of the polygon, the more
it resembled the shape of the circle.
Specifically the perimeter of the polygon
approximates the circumference of the circle.
So lets derive a formula to calculate the
perimeter of the inscribed polygon with an
arbitrary number of sides.
We know that the radius is 1/2.
And we'll call the center of the polygon point
A, with points B and C as two verticies.
Next with line BC and point A, we construct
line AD such that it is perpendicular to line
BC.
And incidentally point D bisects line BC.
So triangle ABD is a right triangle with sides
a, b, and c and an angle called theta.
Line DC also has length "a."
Now in order to find the perimeter of an equilateral
polygon we must find the length of of one
of its sides.
Using trigonometry, we know that sin of (theta)
is equal to the opposite side over hypothenuse.
Looking at our right triangle we can see that
the side opposite angle theta is a, and the
hypothenuse is 1/2.
So a over 1/2 simplifies to two a.
Next looking at one of the sides of the polygon,
we can see that it is actually equal to a
+ a.
Which simplifies to 2a.
Therefore the length of a side of the polygon
is equal to sin of theta.
The perimeter of an equilateral polygon is
defined by the number of sides times the length
of a side.
We will simplify by replacing the number of
sides with n, so the perimeter is equal to
n times side.
But we know that side is equal to sin of theta.
Therefore the perimeter is equal to n times
sin of theta.
Finally we must find what angle theta is equal
to.
If we connect all the vertices of the polygon
to the center,
we can see that each individual angle is equal
to 360 degrees divided by the number of sides
of the polygon.
And theta is just equal to 1/2 of one of those
angles.
So theta is equal to 180 degrees over n.
Therefore the perimeter of the polygon is
equal to n times sine of (180 degrees / n)
So looking back at our circle with a circumference
equal to pi, we can understand why increasing
the value of n approximates the value of pi,
because the perimeter of the inscribed polygon
with more sides, approximates the circumference
of the circle.
