Polygon is a closed plane figure bounded by straight lines. There are two basic types of polygons, a convex and a concave polygon. Polygon is said to be convex if no side when extended will pass inside the polygon, otherwise it is concave.

Name of Polygons
n = number of sides of polygon.

n Name
1 Monogon, Henagon (cannot exist)
2 Digon (cannot exist)
3 Triangle, Trigon
5 Pentagon
6 Hexagon
7 Heptagon, Septagon
8 Octagon
9 Nonagon, Enneagon
10 Decagon
11 Undecagon, Hendecagon
12 Dodecagon, Duodecagon
13 Tridecagon, Triskaidecagon
20 Icosagon
30 Triacontagon
40 Tetracontagon
50 Pentacontagon
70 Heptacontagon
80 Octacontagon
90 Enneacontagon
100 Hectogon
1000 Chilliagon
10 000 Myriagon
1 000 000 Megagon

The following are true for convex polygon

1. The sum of the angles of polygon of n sides is 180°(n - 2) right angles.
2. The exterior angles of a polygon are together equal to 4 right angles.

Formulas for convex polygon

Sum of interior angles
$\Sigma \beta = 180^\circ (n - 2)$

Sum of exterior angles
$\Sigma \alpha = 360^\circ$

Number of Diagonals
$D = \dfrac{n}{2}(n - 3)$

## The Regular Polygon

Rhombus is a quadrilateral with all sides equal (equilateral). Rectangle is a quadrilateral with all included angles are equal (equiangular). Square is both equilateral and equiangular, thus square is a regular polygon. Regular polygons are polygons with all sides equal and all included angles equal. Meaning, regular polygons are both equilateral and equiangular.

Properties of regular polygons

1. The center of the circumscribing circle, the center of inscribed circle, and the center of polygon itself are coincidence.
2. All sides of regular polygon are equal in length; it is denoted by x in the figure.
3. All included angles are equal; it is denoted by β.
4. All external angles α, are equal.
5. Central angles of each segment are equal; it is denoted by θ.
6. The apothem is the radius of the inscribed circle, r.
7. The number of sides is equal to the number of vertices, both are denoted by n.
8. Diagonals that pass through the center has length equal to the diameter of the circumscribing circle.
9. The triangular segment with area denoted as A1 is an isosceles triangle. The length of the two equal sides of this triangle is the radius of the circumscribing circle and the altitude of this triangle is the radius of the inscribed circle.

Formulas for a Regular Polygon

Area of one segment, A1
$A_1 = \frac{1}{2} xr$

$A_1 = \frac{1}{2}R^2 \sin \theta$

Total area, A
$A = nA_1$

$A = \dfrac{n}{2}xr$

$A = \dfrac{n}{2}R^2 \sin \theta$

Perimeter, P
$P = nx$

Central angle, θ
$\theta = \dfrac{360^\circ}{n}$

Exterior angle, α
$\alpha = \theta$

$\alpha = \dfrac{360^\circ}{n}$

Interior angle, β
$\beta = 180^\circ - \alpha$

$\beta = 180^\circ \left( \dfrac{n - 2}{n} \right)$

Where
A1 = area of one segment
A = total area
x = length of side
r = radius of the inscribed circle (apothem)
R = radius of the circumscribing circle
n = number of sides
θ = central angle
α = exterior angle
β = interior angle

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