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.
- This article explains the key methods for calculating the area of a circle, including the standard formula A = pi times r squared, the classical Rearrangement proof attributed to Archimedes, and modern calculus-based proofs
- All mathematical content is based on established geometry and has been reviewed for accuracy.
What Are the Methods of Calculating the Area of a Circle?
There are various methods to calculate the area of a circle. The standard formula is A = πr², where r is the radius and π is approximately 3.14159. One historic method was given by Archimedes, who approximated the area by inscribing and circumscribing regular polygons around the circle. Modern calculus provides additional rigorous proofs.
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.
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What Is the Old Proof of the Area of a Circle?
The classic old proof uses the Rearrangement method by inscribing regular polygons. If a polygon has 2n sides inscribed in a circle, it can be rearranged into a near-parallelogram. As the number of sides increases, the base of this parallelogram approaches half the circle circumference and the height approaches the radius, giving A = πr².
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.
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What Are the Modern Proofs of the Area of a Circle?
Modern proofs of the area of a circle use calculus and rigorous mathematical definitions. The Onion proof sums up the area incrementally by partitioning the disk into thin concentric rings. Other proofs use integration in polar or Cartesian coordinates. These approaches are considered more rigorous than the classical geometric arguments used by Archimedes and earlier mathematicians.
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.
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How Did Archimedes Calculate the Area of a Circle?
Archimedes calculated the area of a circle by stopping at a polygon of 96 sides, obtaining the approximation that π is between 3 + 10/71 and 3 + 1/7. The formula A = πr² results from this exhaustion method. Later mathematicians continued Archimedes' approach using polygons with more sides to achieve greater accuracy for π.
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.