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In Euclidean geometry, a regular polygon is a polygon that is direct equiangular ... For regular polygons with side s = 1, circumradius R = 1, or apothem a = 1, ...
More generally, an n-sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case n = 4, a cyclic quadrilateral. All rectangles, isosceles trapezoids, right kites, and regular polygons are cyclic, but not every polygon is.
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
The regular octagon, in terms of the side length a, has three different types of diagonals: Short diagonal; Medium diagonal (also called span or height), which is twice the length of the inradius; Long diagonal, which is twice the length of the circumradius. The formula for each of them follows from the basic principles of geometry.
Area A of a regular convex polygon with n sides and side length s: = Inradius r of a regular convex polygon with n sides and side length s: = Circumradius R of a regular convex polygon with n sides and side length s:
Side (), circumradius (), inscribed circle radius (), height (+), width/diagonal ()A regular pentagon has Schläfli symbol {5} and interior angles of 108°.. A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°).
Such a circle is said to circumscribe the points or a polygon formed from them; such a polygon is said to be inscribed in the circle. Circumcircle, the circumscribed circle of a triangle, which always exists for a given triangle. Cyclic polygon, a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic ...
In a regular polygon, the incircle and the circumcircle are concentric—that is, they share a common center, which is also the center of the regular polygon, so the distance between the incenter and circumcenter is always zero. The radius of the inscribed circle is the apothem (the shortest distance from the center to the boundary of the ...