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The sum of the squared distances from the vertices of a regular n-gon to any point on its circumcircle equals 2nR 2 where R is the circumradius. [4]: p.73 The sum of the squared distances from the midpoints of the sides of a regular n-gon to any point on the circumcircle is 2nR 2 − 1 / 4 ns 2, where s is the side length and R is the ...
Let one n-gon be inscribed in a circle, and let another n-gon be tangential to that circle at the vertices of the first n-gon. Then from any point P on the circle, the product of the perpendicular distances from P to the sides of the first n-gon equals the product of the perpendicular distances from P to the sides of the second n-gon. [13]
Given a circle, let u n be the perimeter of an inscribed regular n-gon, and let U n be the perimeter of a circumscribed regular n-gon. Then u n and U n are lower and upper bounds for the circumference of the circle that become sharper and sharper as n increases, and their average ( u n + U n )/2 is an especially good approximation to the ...
If n = pq with p = 2 or p and q coprime, an n-gon can be constructed from a p-gon and a q-gon. If p = 2, draw a q-gon and bisect one of its central angles. From this, a 2q-gon can be constructed. If p > 2, inscribe a p-gon and a q-gon in the same circle in such a way that they share a
The area of a regular 65537-gon is (with t = edge length) A = 65537 4 t 2 cot π 65537 {\displaystyle A={\frac {65537}{4}}t^{2}\cot {\frac {\pi }{65537}}} A whole regular 65537-gon is not visually discernible from a circle , and its perimeter differs from that of the circumscribed circle by about 15 parts per billion .
In geometry, a polygon (/ ˈ p ɒ l ɪ ɡ ɒ n /) is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its edges or sides. The points where two edges meet are the polygon's vertices or corners. An n-gon is a polygon with n sides; for example, a triangle is a 3 ...
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The edges of the two regular polygons are assumed to have the same length. A point z attached rigidly to the n-gon traces out an arch consisting of n circular arcs before repeating the pattern periodically. This curve is called a trochogon — an epitrochogon if the n-gon rolls outside the m-gon, and a hypotrochogon if it rolls inside the m-gon ...