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The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
Gauss's circle problem asks how many points there are inside this circle of the form (,) where and are both integers. Since the equation of this circle is given in Cartesian coordinates by x 2 + y 2 = r 2 {\displaystyle x^{2}+y^{2}=r^{2}} , the question is equivalently asking how many pairs of integers m and n there are such that
Following Archimedes' argument in The Measurement of a Circle (c. 260 BCE), 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.
The circular arc is said to subtend the angle, known as the central angle, at the centre of the circle. The angle subtended by a complete circle at its centre is a complete angle, which measures 2 π radians, 360 degrees, or one turn.
A circle containing one acre is cut by another whose center is on the circumference of the given circle, and the area common to both is one-half acre. Find the radius of the cutting circle. The solutions in both cases are non-trivial but yield to straightforward application of trigonometry, analytical geometry or integral calculus.
The problem of finding the area under an arbitrary curve, now known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. [10] Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass ...