Properties of Polygons Study Guide
Practice problems for these concepts can be found at: Properties of Polygons Practice Problems
Introduction to Properties of Polygons
In this lesson, you will learn how to determine whether a figure is a polygon. You will also learn how to identify concave and convex polygons. You will learn how to classify polygons by their sides and how to find the measures of their interior and exterior angles.
The word polygon comes from Greek words meaning "many angled." A polygon is a closed plane figure formed by line segments. The line segments are called sides that intersect only at their endpoints, which are called vertex points.
A polygon is convex when no segment connecting two vertices (vertex points) contains points outside the polygon. In other words, if you wrapped a rubber band around a convex polygon, it would fit snugly without gaps. A concave polygon has at least one place that "caves in."
Parts of a Polygon
Two sides of a polygon that intersect are called consecutive or adjacent sides. The endpoints of a side are called consecutive or adjacent vertices. The segment that connects two nonconsecutive vertices is called a diagonal of the polygon.
When naming a polygon, you name its consecutive vertices in clockwise or counterclockwise order. Here are a few of the ways to name the following polygon: ABCDE, DEABC, or EDCBA.
Although you can start at any vertex, you cannot skip around when you name a polygon. For example, you cannot name the polygon BDEAC, ECBAD, or ACEDB.
Polygons are classified by their number of sides.
Finding the Measure of Interior Angles
There are two theorems that you can use to find the measure of interior angles of a convex polygon. One theorem works only for triangles. The other theorem works for all convex polygons, including triangles. Let's take a look at the theorem for triangles first.
To illustrate this, cut a triangle from a piece of paper. Tear off the three angles or points of the triangle. Without overlapping the edges, put the vertex points together. They will form a straight line or straight angle. Remember that a straight angle is 180°; therefore, the three angles of a triangle add up to 180°, as shown in the following figures.
You can find the sum of the interior angles of a convex polygon if you know how many sides the polygon has. Look at these figures. Do you see a pattern?
The diagram suggests that polygons can be divided into triangles. Since each triangle has 180°,multiply the number of triangles by 180 to get the sum of the interior angles.
Look for a pattern in the number of sides a polygon has and the number of triangles drawn from one vertex point. You will always have two fewer triangles than the number of sides of the polygon. You can write this as a general statement with the letter n representing the number of sides of the polygon.
Find the sum of the interior angles of a polygon with 12 sides.
n = 12
S = 180(n – 2)
S = 180(12 – 2)
S = 180(10)
S = 1,800
Therefore, the sum of the interior angles of a 12-sided polygon is 1,800°.
Finding the Measure of Interior Angles of Regular Polygons
A regular polygon is any polygon whose interior angles all have the same angle measurement. Recall that the formula to calculate the sum, S, of the interior angles of a polygon with n sides is S = 180(n–2). In order to calculate the measure of the interior angle of a regular polygon, divide the sum, S, by the number of sides, n.
Find the measure of an interior angle of a regular pentagon.
Finding the Measure of Exterior Angles
Use this theorem to find the measure of exterior angles of a convex polygon.
To illustrate this theorem, picture yourself walking alongside a polygon. As you reach each vertex point, you will turn the number of degrees in the exterior angle. When you return to your starting point, you will have rotated 360°.
This figure shows this theorem using a pentagon. Do you see that this would be true for all polygons as stated in the theorem?
Practice problems for these concepts can be found at: Properties of Polygons Practice Questions.
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