deriving the formula for the area of a circle without knowing pi and by using calculus

Say we have a square inscribed in a circle. The diagonal of the square will equal the diameter (twice the radius or 2*r) of the circle. For any square, the length of the diagonal is the length of a side times the squared root of two. Thus, if we let x be the side length, x*sqrt(2) = 2*r and so x = r*sqrt(2). Thus, the area of the square is :

(r*sqrt(2))^2 = 2*r^2 

That's close to the actual area of a circle. If we repeat this process with regular polygons with more sides, the answer should converge on the correct formula.

OK, let's repeat with a pentagon. It can be cut up into 5 isosceles triangles which each have 2 sides of length r. Using the formula for the sum of interior angles (n-2)*180, we get each angle of the pentagon has 108 degrees ((5-2)*180/5 = 108). That means that 2 of the angles in each triangle are 108/2 = 54 degrees. If we draw the altitude of each triangle, we get right triangles where the hypotenuse is r and the two acute angles are 54 and 36 degrees. The cosine of 36 degrees equals the altitude divided by r and the sine of 36 equals half the pentagon side length divided by r. So r*cos(36) is the altitude or height and 2*r*sin(36) is the base of each isosceles triangle. Thus the area of the pentagon is 5 times the area of each isosceles triangle or 5*(1/2)*(2*r*sin(36))(r*cos(36)). This is about 2.37*r^2, so a bit closer to the true formula.

We can generalize this process for polygons with n sides. The area of such a polygon is 

n*1/2*2*r*sin(x)*r*cos(x) = n*sin(x)*cos(x)*r^2

where x = 90-(n-2)*180/(2*n)

As n increases, x decreases. For n = 10, we get x = 18 degrees and an area of about 2.94*r^2

For n = 100, we get x = 1.8 and an area of about 3.14*r^2. 

For n = 1000, we x = 0.18 and an area of about 3.1415*r^2

This is similar to the idea of realizing that the base of each triangle approaches 2*pi*r/n and the height approaches r as n in increases. The area formula then becomes the limit as n increases of n*1/2*base*height = n*1/2*(2*pi*r/n)*r = pi*r^2

It's a bit easier with calculus. Take a circle of radius R and cut it into many thin rings. Each ring can be cut and unrolled to make rectangles of length 2*pi*r and thickness of dr. Integrate from 2*pi*r dr from 0 to R and you get pi*R^2 as the area of the circle. 

There are often many paths to the top of a mountain.