How to Find the Area of a Circle by Inductive Reasoning
In inductive reasoning, a trend is observed, it is assumed that it is true and that it will continue to be so, or taking on a format that allows it to be predicted based on what has been observed and something is stated in general.
We will apply this methodology to obtain the formula for the area A꜀ of a circle.
In the flat figure (imagine it as a thin pizza on a table) above we can identify the elements: the segment drawn between the points ρ and α indicates that it is a radius of length R; the curved and closed line c is the circle centered at the point ρ. Measuring the region inside the curve is therefore our objective.
Let’s establish a second ray in the figure with its extremity at the point β, so that we obtain a sub region (a slice of the pie) called circular sector c′. Now let’s repeat this procedure in order to have n number of sectors (the circle will contain n identical slices)
So the area of the circle will be given by multiplication:
Now what we need is to determine the area of the circular sector c′ of our figure. To do this, we will make an approximation considering that the circular sector approaches the figure of an isosceles triangle (two adjacent sides of equal length) underlying it, as long as the base of this triangle corresponds to the length of the arc of the circular sector c′.
Thus we can say that the area of the circular sector c′ has a value close to that of its underlying triangle:
Regarding the circle c we can write
Using inductive thinking, we can consider that when we divide the circle into infinite sectors N, the sector-triangle approximation will become closer so that we can write
Continuing, we can develop this expression using the known formula for the area of the triangle.
and then
but the base of our triangle will correspond in length to one of the N divisions of the circle’s perimeter, that is:
and therefore,
which, simply put, results in
