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The regular decagon has Dih 10 symmetry, order 20. There are 3 subgroup dihedral symmetries: Dih 5, Dih 2, and Dih 1, and 4 cyclic group symmetries: Z 10, Z 5, Z 2, and Z 1. These 8 symmetries can be seen in 10 distinct symmetries on the decagon, a larger number because the lines of reflections can either pass through vertices or edges.
In terms of the circumradius R, the area is: [1] = = The span S of the dodecagon is the distance between two parallel sides and is equal to twice the apothem. A simple formula for area (given side length and span) 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 segment from the center to the midpoint of one of its sides.
Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities; List of volume formulas – Quantity of three-dimensional space
The hydraulic diameter is the equivalent circular configuration with the same circumference as the wetted perimeter. The area of a circle of radius R is . Given the area of a non-circular object A, one can calculate its area-equivalent radius by setting = or, alternatively:
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.
Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N 3 and N 5; then if ordinates N 3 P 3, N 5 P 5 are drawn to the circle, the arcs AP 3, AP 5 will be 3/17 and 5/17 of the circumference." The point N 3 is very close to the center point of Thales' theorem over AF.
Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols. These properties apply to all regular polygons, whether convex or star: . A regular n-sided polygon has rotational symmetry of order n.