All three can be done by realizing that all circles are the same shape so "similar". A little more technical way of saying that is that since circumference is a distance measurement, just as diameter is, circumference is proportional to diameter- circumference is a multiple of diameter- and we define to be that multiple. One way to prove the area formula, that goes back to Archimedes, is to divide the circle into n wedges with equal interior angle, then "alternate" them so that the first has its "point" downward, the next upward, the next downward and so on. That gives a figure that "looks like" a rectangle with a "wavy" edge. That rectangle would have one side of length r and the other half the circumference, , and so area . Take the limit as n goes to infinity and the thing that "looks like" a rectangle becomes a rectangle.How to prove that the circumference of a circle = 2*pi*r
How to prove area of circle = pi*r^2
Of course, a "modern" way to do this would be to use an arclength integral for the circumference and an area integral for the area.
If you let x= r cos(t), y= r sin(t), with t from 0 to then so those are parametric equations for the circle of radius r. The circumference is given by and the area is given by . Let x=r sin(t) so that dx=r cos(t)dt and that becomes . so we can write that as
The "sin(2t)" evaluated at and is 0. That leaves .
One might argue that invalid to use sine and cosine in parametric equations or to use the sine substitution for the area integral because those functions are themselves defined in terms of distance around a circle but that is not true. It is perfectly valid to define "sin(x)" as the power series and define "cos(x)" as the power series . Or, equivalently, define sin(x) to be the solution to the differential equation y''= -y with initial values y(0)= 0, y'(0)= 1 and define cos(x) to be the solution to the differential equation y''= -y with initial values y(0)= 1, y'(0)= 0.