It is one of the most commonly used formulae in mathematics. We all know that the are of a circle is pi*r^2. But, why is it pi*r^2?
Imagine a circle with multiple spokes with each spoke length being ‘r’, and the distance between the spokes is ‘x’. It becomes a tiny triangle with ‘r’ as the height and ‘x’ as the base. In that case, the area of that small triangle will be ½*x*r = rx/2 (shown below)
Now, imagine how many such tiny triangles will be there, the base is of length ‘x’ and is part of the circumference 2*pi*r. So, the number of triangles = number of bases that cover 2*pi*r = 2*pi*r/x
So, the total area of the circle will be the sum of the areas of these tiny triangles = rx/2 * (2*pi*r/x) = pi*r^2 (as shown below)
In calculus, all methods to calculate area or volume involve dividing and approximating with the closest shapes and integrating those areas over a numerous times.
Similarly, even if you want to find out the area of any irregular shape, you would break it up into tiny portions of a shape that you know the area for and then approximate it. This is the fundamental intuition of calculus.
So, in this post, we saw how the area of a circle is approximated with tiny right-angled triangles of height ‘r’ and base ‘x’. Eventually, it shows the area of the circle is pi*r^2.
Hope this is useful, thank you.
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