How was the formula for the area of a circle determined?

While Archimedes calculated the value of pi and the area of a circle through his elegant use of polygons, a more straight forward, and, as elegant, relationship was undoubtly known between the area of a circle and the area of a parallegram.

The Babylonians (2000 BC) knew that there was a relationship between the Circumference of a Circle and its Diameter, i.e., X = Circumference / Diameter. This relationship was defined by the symbol in the 18 th century AD so that:

= Circumference (C) / Diameter (D) or
C = 2r, where r is the radius of the Circle (1/2 the Diameter)
Its value was estimated (by direct measurement) by the Babylonians to be: = 3.125
The value of used today is: 3.142

If a circle is divided into equal sections (wedges) and these sections (wedges) are then cut out and arranged end-to-end, a parallelagram is formed with a Width equal to the Radius (r) of the circle, and a Length (top and bottom sides each) equal to one half (1/2) of the Circumference of the circle or 1/2(2r) or : r

Image Source

Since the area of a parallelagram is equal to Length (L) X Width (W), this becomes Area = L x W = (r) x r = r2. This is the formula for the calculation of the area circle:
Area of a Circle = r2.