The circle, in Euclidean geometry, is a projection of one end of a line (radius) that is axially pivoted along a point (centre). This shape is widely employed; a particularly significant example will be the shape of a wheel. Determination of the amount of material needed to make cylindrical objects will require calculating the volume, which means the area of circle will have to be obtained.
Now that the need to have the area of a circle is established, we now see how mathematicians have strained themselves to approaching the actual area of a circle.
Archimedes started the ball rolling by approximating the shape of a circle by plotting out regular polygons, for the circle can be taken as an infinitesimal polygon. This method when the sides of the plotted polygon become larger. With a circle circumscribed and inscribed into a plotted polygon, he was able to obtain a range of values for the area of the circle. Archimedes extracted an average from the two extreme bounds of the inequality and made a verdict to the determination of the area of the circle. This method proved to be ingenious, albeit inefficient and archaic.
The ratio of the circle's circumference and its diameter indicates the proportion of these values and are exploited by mathematicians to obtain the area of the circle. The simplest way to find an area of a circle is a modest equation which goes by:
area = pi x r squared. Where pi is 3.41592654.., and r is half of the diameter
This suggests that when a circle is inscribed into a square, that area of circle is approximately 79% of the area of the square.
The Onion proof is an integral calculus demonstration to the derivation of the equation mentioned above. The idea is that a circle is imagined to be a sphere with an infinite number of shells. The shell does not have thickness (infinitely thin), hence the circumference is the appropriate representation of the length. Integrating the domain from the centre to the end of the radius yields the area of the circle, as a definite integration process combines the infinitesimal layers of the circle to form a definite value. This property is known as convergence.
One last method of approximation is purely for the curious minds. The Monte Carlo method utilizes the depiction of a square with a circle inscribed into it. The idea is to throw darts randomly from the top of the picture and count where these darts land. The ratio of the number of darts fell to the circle and the number of darts fell to the square is multiplied by the area of square to give the approximate area of the inscribed circle.
These are the few methods devised by mathematicians to determine the area of the circle. Many more methods were discovered and are more mathematically by definition. They can be accessed from the reference section of this article.
Archimedes (c. 260 BCE), "Measurement of a circle", in T.L.Heath (trans.), The Works of Archimedes, Dover, 2002, pp. 9193, ISBN 978-0-486-42084-4
Beckmann, Petr (1976), A History of Pi, St, Martin's Griffin, ISBN 978-0-312-38185-1