The area of a circle is the region occupied by the circle in a two-dimensional plane. The area of a circle is given by the formula πr². π is given by a Greek letter, called as ‘pi’, which represents a constant which is equal to 3.1415926 approximately. This pi is equal to the ratio of the circumference of any circle to its diameter. r in the formula is the radius of the circle which is half the diameter of a circle. 

Methods of Calculating Area Of Circle

There are various methods and proofs to obtain this formula finally. One such method was given by Archimedes which views circle as a sequence of regular polygons. The area of a regular polygon is the product of half its perimeter and distance from the center to the sides, that is, 12*2r*r. This gives the area of circle. Since the time of establishing this equation, there were various arguments made to prove the equation. The most famous was the Archimedes method of exhaustion under various degrees of mathematical rigour.

The original proof of Archimedes is considered not rigorous by modern standards. Following Archimedes’ argument in ‘The Measurement of a Circle‘, compare the area enclosed by a circle to a right triangle whose base has the length of the circle’s circumference and whose height equals the circle’s radius. If the area of the circle is not equal to that of the triangle, then it must be either greater or less. We eliminate each of these by contradiction, leaving equality as the only possibility. We similarly use regular polygons. 

Check Out: Area of an Equilateral Triangle

Old Proof Of Area Of Circle

The other proof used is the Rearrangement proof by inscribing the hexagons. Simply put, if a polygon is with 2n sides then the parallelogram will have a base of length ns and a height h.

As the number of sides increases, the length of the parallelogram base approaches half the circle circumference and its height approaches half the circle radius. By this, the parallelogram turns into a rectangle with dimensions of width πr and height r.

Check Out: Area of Rectangle

Modern Proof Of Area Of Circle

The various modern proofs given are plenty in number by using calculus. There are huge amounts of definitions available for constant pi. The most conventional definition in pre-calculus geometry is defined as the ratio of the circumference to the diameter of the circle. π=CD. This is not a suitable definition anymore. The modern definition is that pi is equal to twice the least positive root of the cosine function which is also equivalent to the half-period of the sine function.

Few of the modern proofs are:

Onion proof: This proof uses calculus. The area is summed up incrementally by partitioning the disk (i.e. onion) into concentric rings like layers of an onion. To this integration is applied. 
Triangle proof: This method proves the area of a circle by unwrapping the concentric circles to straight strips forming a right-angled triangle with height r and 2πr as its base. 
Semicircle proof: The semi-circle of a circle is computed by the integration and there is also trigonometric substitution to prove the answer.

Check Out: AREA OF RHOMBUS

Archimedes’ Calculation

Archimedes’ calculations for approximation were heavy and laborious and he stopped with a polygon of 96 sides. The new faster methods were found by other mathematicians. Few such methods are Archimedes doubling method, The Snell-Huygens refinement, Dart approximation, finite arrangement, Non-Euclidean Circles, etc.

The above information is based on the area of circle and the history of developing and proving it that we use for daily geometrical mathematics.

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