Properties of Polygons (Sides, Angles and Diagonals)

Angles

Partitioning an n-gon into n − 2 triangles

Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:

• Interior angle – The sum of the interior angles of a simple n-gon is (n − 2)π radians or (n − 2) × 180 degrees. This is because any simple n-gon ( having n sides ) can be considered to be made up of (n − 2) triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is $\left(1-{\frac {2}{n}}\right)\pi$ radians or $180-{\frac {360}{n}}$ degrees. The interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular ${\frac {p}{q}}$-gon (a p-gon with central density q), each interior angle is ${\frac {\pi (p-2q)}{p}}$ radians or ${\frac {180(p-2q)}{p}}$ degrees.

• Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight" or antiparallelogram, where d is the density or turning number of the polygon. See also orbit (dynamics).

Area

Coordinates of a non-convex pentagon.

In this section, the vertices of the polygon under consideration are taken to be $(x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})$ in order. For convenience in some formulas, the notation $(x_{n},y_{n})=(x_{0},y_{0})$ will also be used.

If the polygon is non-self-intersecting (that is, simple), the signed area is

$A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}\quad {\text{and}}\quad y_{n}=y_{0}$,

or, using determinants

$16A^{2}=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}{\begin{vmatrix}Q_{i,j}&Q_{i,j+1}\\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}}$,

where $Q_{i,j}$ is the squared distance between $(x_{i},y_{i})$ and $(x_{j},y_{j})$.

The signed area depends on the ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive x-axis to the positive y-axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in absolute value. This is commonly called the shoelace formula or Surveyor's formula.

The area A of a simple polygon can also be computed if the lengths of the sides, a1, a2, ..., an and the exterior angles, θ1, θ2, ..., θn are known, from:

${\begin{aligned}A={\frac {1}{2}}(a_{1}[a_{2}\sin(\theta _{1})+a_{3}\sin(\theta _{1}+\theta _{2})+\cdots +a_{n-1}\sin(\theta _{1}+\theta _{2}+\cdots +\theta _{n-2})]\\{}+a_{2}[a_{3}\sin(\theta _{2})+a_{4}\sin(\theta _{2}+\theta _{3})+\cdots +a_{n-1}\sin(\theta _{2}+\cdots +\theta _{n-2})]\\{}+\cdots +a_{n-2}[a_{n-1}\sin(\theta _{n-2})])\end{aligned}}$.

The formula was described by Lopshits in 1963.

If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.

In every polygon with perimeter p and area A , the isoperimetric inequality $p^{2}>4\pi A$ holds.

For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon.

The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides do determine the area. Of all n-gons with given side lengths, the one with the largest area is cyclic. Of all n-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).

Regular polygons

Many specialized formulas apply to the areas of regular polygons.

The area of a regular polygon is given in terms of the radius r of its inscribed circle and its perimeter p by

$A={\frac {1}{2}}\cdot p\cdot r.$

This radius is also termed its apothem and is often represented as a.

The area of a regular n-gon in terms of the radius R of its circumscribed circle can be expressed trigonometrically as:

$A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}}$

The area of a regular n-gon inscribed in a unit-radius circle, with side s and interior angle $\alpha$, can also be expressed trigonometrically as:

$A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}={\frac {ns^{2}}{4}}\cot {\frac {\alpha }{n-2}}=n\cdot \sin {\frac {\alpha }{n-2}}\cdot \cos {\frac {\alpha }{n-2}}.$

Self-intersecting

The area of a self-intersecting polygon can be defined in two different ways, giving different answers:

• Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the density of the region. For example, the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure.

• Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon or to the area of one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral, it is treated as two simple triangles.

Centroid

Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are

$C_{x}={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i})$,

$C_{y}={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i})$.

In these formulas, the signed value of area A must be used.

For triangles (n = 3), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for n > 3. The centroid of the vertex set of a polygon with n vertices has the coordinates

$c_{x}={\frac {1}{n}}\sum _{i=0}^{n-1}x_{i}$,

$c_{y}={\frac {1}{n}}\sum _{i=0}^{n-1}y_{i}$.