Once has been defined as the ratio circumference to diameter, the area of a circle must be .
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.
Abstract algebra, the un-codified implications of which need to be fully understood at school to gain good grades in exams. School algebra and numeracy is all about persuading pupils that the numbers we have work by these rules, and not by the various coping patterns they learned earlier in their education.
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.
Carr’s Synopsis – 19th century text for self taught geniuses – marks out this A-star topic for I/GCSE maths hopefuls.
Added as complement to last post.
This 7th century visualisation of quadratic algebra, thanks to Islamic scholars, is a great place to start the academic year in mathematics.
Problems with straight line graphs
Rules of Indices provide an excellent first look at a developing mathematical structure. You start with a definition which provides a nice rule. You then generalise, trying to preserve the rule. What you get is a marvellous structure which is all a consequence of the first definition.