Area of a circle – proof by exhuastion

Once \pi has been defined as the ratio circumference to diameter, the area of a circle must be \pi r^{2}.

A proof relies on an infinite, limiting process which paves the way to some calculus-like ideas.

A circle is cut up into 6 sectors which are then rearranged into a near rectangle,

What doesn’t look too close to a rectangle at 6 sectors looks better at 26:

As the number of sectors becomes very big the shape becomes indistinguishable from a rectangle and the argument is complete.

I think that Archimedes used such arguments, or, proof by exhaustion, in some of his solid geometry work.

Carr’s synopsis reveals much about cyclic quadrilaterals and the cosine rule.  This Victorian text book is packed with sparse facts.  Genius Ramanujan worked through these proving each one to himself.  Modern students would extend their skills by doing the same.  I have added some connections and associations to this sequence of facts to get you started.  Should be within reach of year 11 pupils.