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Apothem of a hexagon Graphs of side, s; apothem, a; and area, A of regular polygons of n sides and circumradius 1, with the base, b of a rectangle with the same area. The green line shows the case n = 6. The apothem (sometimes abbreviated as apo [1]) of a regular polygon is a line
In geometry, a polygon is traditionally a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain. These segments are called its edges or sides , and the points where two of the edges meet are the polygon's vertices (singular: vertex) or corners .
For a regular polygon with 10,000 sides (a myriagon) the internal angle is 179.964°. As the number of sides increases, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle.
The apothem is half the cotangent of /, and the area of each of the 14 small triangles is one-fourth of the apothem. The area of a regular heptagon inscribed in a circle of radius R is 7 R 2 2 sin 2 π 7 , {\displaystyle {\tfrac {7R^{2}}{2}}\sin {\tfrac {2\pi }{7}},} while the area of the circle itself is π R 2 ; {\displaystyle \pi R^{2 ...
A regular polygon with n sides can be constructed with ruler, compass, and angle trisector if and only if =, where r, s, k ≥ 0 and where the p i are distinct Pierpont primes greater than 3 (primes of the form +). [8]: Thm. 2 These polygons are exactly the regular polygons that can be constructed with Conic section, and the regular polygons ...
As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, [11] which comes to π multiplied by the radius squared: =.
The arc length, from the familiar geometry of a circle, is s = θ R {\displaystyle s={\theta }R} The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of θ {\displaystyle \theta } ):
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. [1]