Regular Polygon

A regular polygon is a polygon that is equiangular and equilateral. A regular polygon has all equal edges and all equal interior angles. Regular polygons may be either convex or star.

Regular Convex Polygon

Properties of Regular Convex Polygon

Symmetry

All axes of symmetry pass through the center of the polygon. For polygons with even number of sides, half of the axes pass through two opposite vertices diagonally. While the other half of axes pass through the midpoints of opposite edges. For polygons with odd number of sides, all the axes of symmetry pass through a vertex and the midpoint of its opposite edge.

Interior Angle and Central Angle

For a regular convex polygon of sides, 𝑛, the sum of all interior angles is equal to (𝑛-2)𝜋. The interior angle, 𝜑, is equal to ((𝑛-2)𝜋)/𝑛=𝜋−2𝜋/𝑛. Therefore, all interior angles of a regular convex polygon should be less than 𝜋 radians or 180°. The sum of all central angles is equal to 2𝜋. The central angle, 𝜃, is equal to 2𝜋/𝑛.

Circumcircle and Incircle

The circumcircle or cirmuscribed circle is a circle that passes through all vertices of the regular convex polygon. The center of the circumcircle is the center of the regular convex polygon. The radius of circumcircle, 𝑟_𝑐, is usually called circumradius. The circumradius is equal to (1/(2sin(𝜃/2)))𝑎=(1/(2sin(𝜋/𝑛)))𝑎, where 𝑎 is the edge of the regular convex polygon. The incircle or inscribed circle is a circle that passes through all midpoints of edges of the regular convex polygon. The center of the incircle is the center of the regular convex polygon. All edges of the regular convex polygon are tangents to the incircle. The radius of incircle, 𝑟_𝑖, is usually called inradius. The inradius is equal to (1/(2tan(𝜃/2)))𝑎=(1/(2tan(𝜋/𝑛)))𝑎, where 𝑎 is the edge of the regular convex polygon. And inradius, 𝑟_𝑖, is also equal to 𝑟_𝑐cos(𝜃/2)=𝑟_𝑐cos(𝜋/𝑛). As increasing the number of edges, 𝑛, the inradius, 𝑟_𝑖, asymptotically tends to the circumradius, 𝑟_𝑐, because angle 𝜃 approaches zero.

Area

The total area, 𝐴, of a regular convex polygon can be divided into 𝑛 identical isosceles triangles. Since the height of these isosceles triangles are perpendicular to the edge, 𝑎, of the regular convex polygon and is equal to the radius of the incircle, that is 𝑟_𝑖. Therefore, the area of each isosceles triangles is 𝑎𝑟_𝑖/2, and the total area of the 𝑛 triangles is 𝑛𝑎𝑟_𝑖/2=𝑛𝑎²4tan(𝜋/𝑛). Or expressed in terms of circumradius, 𝑟_𝑐, that is 𝐴=𝑛𝑎𝑟_𝑖/2=𝑛(2𝑟_𝑐sin(𝜋/𝑛))(𝑟_𝑐cos(𝜋/𝑛))/2=𝑛(𝑟_𝑐)²sin(2𝜋/𝑛)/2.

Perimeter

The perimeter, 𝑃, of a regular convex polygon is equal to the sum of the lengths of all edges, that is 𝑃=𝑛𝑎.

Bounding Box

The bounding box of a planar shape is the smallest rectangle that encloses the shape completely. The dimensions of a bounding box depends on the number of edges of a regular convex polygon.

Height

The hegiht of the bounding box is defined by taking the stable resting position of the regular convex polygon. If the number of edges, 𝑛, of a regular convex polygon is even, the height, ℎ, of the boundary box is the distance between two opposite edge midpoints. If the number of edges, 𝑛, of a regular convex polygon is old, the height, ℎ, of the boundary box is the distance between a vertex and the midpoint of the corresponding opposite edges. Therefore, ℎ=2𝑟_𝑖 for 𝑛 is even and ℎ=𝑟_𝑖+𝑟_𝑐 for 𝑛 is old.

Width

The width, 𝑤, of the bounding box can be divided into three cases. If 𝑛 is even and 𝑛/2 is even also, the width, 𝑤, is equals to 2𝑟_𝑖. If 𝑛 is even and 𝑛/2 is odd, the width, 𝑤, is equals to 2𝑟_𝑐. If 𝑛 is odd, the width, 𝑤, is equals to 2𝑟_𝑐sin((𝑛-1)𝜋/2𝑛).

Source and Reference

• https://en.wikipedia.org/wiki/Polygon
• https://en.wikipedia.org/wiki/Regular_polygon