Understanding why πr^2 works for different area calculations

[paulb203]

TL;DR Summary

A kind of personal 'proof' to help me grasp πr^2

I know there are videos etc explaining why but I thought I would try to find a way to understand this myself.

Imagine a version of πr^2, but instead of being for the area of a circle, it’s for the area of a square.
Call it sqi ar ^2
sqi = the ratio of the ‘diameter’ of the square to the square’s ‘circumference’ (the ‘diameter’ of the square being a vertical or horizontal line through the square’s centre, and the ‘circumference’ being its perimeter).
So, sqi=4 (like π=3.14...)
ar = the ‘radius’ of the square (half it’s ‘diameter’, just like with a circe and it’s radius)

Now take a square 4 cm by 4cm
And apply sqi ar ^2
Which gives us 4x2^2
Which = 16cm^2
Which matches with the 16cm^2 we would get from the conventional way of finding the area of the square.

This helps make sense of πr^2, for me at least
Any thoughts?

 

[scottdave]

That is a start in the understanding. Of course, one example does not make a proof. Try some other examples, such as regular hexagon, octagon, etc.

See how the ratio changes as the limit of number of sides increases.

 

[FactChecker]

The coincidence you are wondering about may be due to the similarity of the figure in the X and Y axis. If you consider the area of a rectangle or an ellipse, the pattern is broken.