jishnu said: I don't know whether it can be proved mathematically that the sides are straight lines

But, when the strips are very thin and as the radius undergoes a gradual decrease in when coming inside, so as the circumstances of inner circles also decreases the sides are going to form straight lines. My knowledge in mathematics is very primitive but, I will try to find a possible solution for this.

In your first drawing in post #9, you are essentially using integration to find the area of a circle, using circular strips, orThe radius of the outer circle is 1, and is centered at the origin. If we divide the circle is divided into thin concentric annuli, each of thickness ##\Delta r##, and of radius r, the area of the annulus is about ##2\pi r \cdot \Delta r##. If ##\Delta r## is reasonably "small" there's not much difference between the inner and outer radii of the annulus. In the drawing I show only one annulus, in blue.To get an approximate value for the circle's area I can add the areas of all the annuli like so:##A \approx \sum_{i = 1}^n 2 \pi r \Delta r##, where r is the radius to some point in the annulus, and ##\Delta r## is the thickness of the annulus, which we can take as ##\frac 1 n##.If we make the annuli thinner (and consequently more of them), the approximation will be better. Where calculus comes in is replacing the sum above (a Riemann sum) by a definite integral: ##\int_0^1 2 \pi r dr##, which turns out, unsurprisingly, to equal ##\pi##. Unfortunately, since the OP wanted to derive the area of a circle without calculus, this solution doesn't meet that requirement.