Area of a Circle - Simple Derivation
There are several ways to derive the formula for the area of a circle. This is my favorite because it's so easy. I call it the "pizza method".
Consider a regular convex polygon with n sides each of length b. It has a "radius" r which is the perpendicular from the center of the polygon to the center of a side.
So the area of one triangular "pizza slice" of the polygon is b*r/2 and thus the total area of the polygon is n*b*r/2
But n*b is the length of the periphery of the polygon, let's call this c for "circumference". So the area of the polygon is c*r/2
So we've found the area of any regular convex polygon, no matter how many sides it has.
But a circle is just a regular convex polygon with an infinite number of sides. Our formula is still valid. So the area of a circle is c*r/2
But by definition pi=c/(2*r) which means c=2*pi*r and so we get the famous formula for the area of a circle pi*r^2
Even though this formula is famous I prefer c*r/2 because it works for a circle and also for any regular convex polygon. So the next time somebody asks you for the area of a circle confuse them and say c*r/2
Content written and posted by Ken Abbott email@example.com