Polygons
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Introduction to "Polygons"
Walking around a city, you can see polygons in buildings, windows, and traffic signs. In general a polygon is a closed plane figure with at least three sides. Those sides intersect only at their endpoints and no adjacent sides are collinear.
Definition of "polygon"
A polygon is a two-dimensional, closed plane figure that has at least three sides, all of which are straight. All polygons have the same definite number of angles and sides.
Regular polygons
A regular polygon is a polygon that is equiangular, equilateral, and the vertices of which are all equidistant from a common center. Simply put, a polygon is considered to be regular if all of its sides have equal length, all of its angles have equal measure, and there exists an imaginary point that is equally distant from each of its corners.
Despite the fact that the uniform side length of any regular polygon has an infinite amount of possible values, the uniform angle measure can be defined by the following formula:
where θ is the angle measure and n is the number of sides the polygon has. This will give the sum of the interior angles of a polygon. It is important to note that this formula is not specific to regular polygons. This formula will give the sum of the interior angles for any polygon.
If a polygon is regular, then the measure of each individual angle is given by:
An example of the use of these two formulas would be finding the measure of each interior angle of a regular pentagon. To find the sum of the interior angles we would use the formula:
Because a pentagon has five sides. This yields an answer of 540 degrees. Dividing this answer by 5 -- because that is the number of sides -- gives an answer of 108 degrees. In an equilateral pentagon, each interior angle has a measure of 108 degrees.
Naming polygons
Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral, and nonagon are exceptions. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.
| Name | Number of Sides | Inscribed Triangles | Angle Sum |
|---|---|---|---|
| monogon | 1 | 0 | 0 |
| digon | 2 | 0 | 0 |
| triangle/trigon | 3 | 1 | 180 |
| quadrilateral/tetragon | 4 | 2 | 360 |
| pentagon | 5 | 3 | 540 |
| hexagon | 6 | 4 | 720 |
| heptagon | 7 | 5 | 900 |
| octagon | 8 | 6 | 1080 |
| nonagon | 9 | 7 | 1260 |
| decagon | 10 | 8 | 1440 |
| hendecagon | 11 | (n - 2) | 180(n - 2) |
| dodecagon | 12 | ||
| tridecagon | 13 | ||
| tetradecagon | 14 | ||
| pentadecagon | 15 | ||
| hexadecagon | 16 | ||
| heptadecagon | 17 | ||
| octadecagon | 18 | ||
| enneadecagon | 19 | ||
| icosagon | 20 | ||
| centagon/hectogon | 100 | ||
| chiliagon | 1000 | ||
| myriagon | 10,000 |
