# Geometry/Chapter 10

## Section 10.1 - 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. Examples of polygons include triangles, trapezoids, and hexagons.

## Section 10.2 - Definition of "polygon"

A polygon is a two-dimensional, closed plane figure that has at least three sides, all of which are straight. Polygon lines cannot intersect other lines of a polygon. All polygons have the same number of angles equal to the number of sides and vice versa.

## Section 10.3 - 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:

$\theta =(n-2)\cdot 180^{\circ }$

where $\theta$  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: $\theta ={\frac {n-2}{n}}\cdot 180^{\circ }$

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:

$\theta =(5-2)\cdot 180^{\circ }$

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.

## Exercises

1. Are all isosceles triangles also equilateral?
2. Is there such thing as a right triangle that is also isosceles?
3. What is the sum of the interior angles in an octagon?
4. What is the measure of an interior angle in a regular hexagon?
5. Draw an equilateral quadrilateral that is also equiangular. Can there be an equilateral quadrilateral that is not equiangular?
6. How many lines of symmetry are there in a regular $n$ -gon?