Polygon; the word means "many angles," but it ignores one attribute: straight sides. Polygons are all around us! Learning about these basic plane shapes helps you understand geometry.
A polygon is a plane figure that closes in a space using only line segments. If it must use only line segments and must close in a space, the polygon with the fewest sides has to be the triangle (three sides and interior angles).
Polygons have no upper limit to the number of sides and interior angles, so mathematicians say n-gon after the familiar names we use for common polygons.
|Polygon Shape||Number of Sides|
Let's take a look at the vast array of shapes that are polygons.
Let's take a closer look at regular polygons. A regular polygon is where all sides are the same length, and all angles are also all the same. Next time you're in a car, take a closer look at a stop sign. A stop sign is an example of a regular polygon with eight equal sides.
Check out these regular polygons. Do you notice how all the sides and angles are the same no matter how you flip it?
The interior angles of polygons are the vertices, or inside corners, created by endpoints of line segments. Connecting all the vertices inside a simple polygon without crossing any lines creates triangles.
Each triangle adds to , so one way to find the sum of interior angles is to count the number of dividing triangles:
Triangle (1 triangle);
Quadrilateral (2 triangles);
Nonagon (7 triangles);
That can get clumsy after a while. You can instead use a formula for the sum of the interior angles. You need to know the number of sides :
Try it yourself, on a triangle:
Now try it on a nonagon, a nine-sided polygon:
All interior angles in a regular polygon are equal (interior angles are congruent). Once you know how to find the sum of interior angles, you can use that to find the measure of any interior angle, , of a regular polygon. Take the same formula and divide by the number of sides:
Let's try it out for an equilateral triangle by plugging in 3 for the number of sides ():
Here is a regular nonagon (9 sides):
The formula can be rearranged like this:
Both versions will give you the same answer. Notice the second part of that version of the formula, . We are going to use that next!
The number of interior angles and exterior angles of any polygon is the same. A triangle has three exterior angles; a square has four; our nonagon has nine.
An exterior angle is an angle created outside the polygon between one side and a line extended from an adjacent side of the polygon. The interior and exterior angles will always add to , so they are supplementary.
On irregular polygons, you have no way of knowing the measure of exterior angles. On regular polygons with sides, you can use a formula to know each of the shape's exterior angles, since they will be congruent:
In an equilateral triangle, the exterior angles come out like this:
Our nonagon does this:
A regular n-gon does this:
Look back at the second version of the formula for finding interior angles of regular polygons, which we wrote as:
Now you can see that the two formulas make sense together because one finds the supplementary angle of the other.
You should now be able to identify and define both a polygon and regular polygon, find the sum of the interior angles of regular and irregular polygons, and find the interior and exterior angles of any regular polygon. You can use formulas to find the sum of interior angles for any polygon and to know the exact measure of both interior and exterior angles for regular polygons.
When you finish viewing the video and reading this lesson, you will learn to:
Get better grades with tutoring from top-rated private tutors. Local and online.
Get better grades with tutoring from top-rated professional tutors. 1-to-1 tailored lessons, flexible scheduling. Get help fast. Want to see the math tutors near you?
Learn faster with a math tutor. Find a tutor locally or online.