Is there a relationship between a polygon's number of sides and its number of vertices and angles?

Answer 1

For a regular polygon, all sides are equal and all angles are equal. Draw a regular hexagon. This website shows you an easy way to draw a hexagon with a compass. http://www.wikihow.com/Draw-a-Hexagon

Draw a line from one vertex to another going thru the center. This should make 2 identical triangles. Do this, until you have 6 triangles. Use a protractor to measure the 6 angles in the center and the base angles. They should each be 60º angles. Two of these adjacent base angles measure 120º. This is the vertex angle. There are 6 vertices, 6 sides, 6 - 60º angles in the center, and 6 - 120º angles at the 6 vertices.

1. If a polygon has n sides, there will be n angles at the center, n vertices and n triangles.

2. Draw the 5 lines connecting each vertex to the center of the polygon making 5 angles.

1. There are 360º around the center point, so 360º ÷ n is the measure of each angle at the center.
2. The angle of the triangle at the center of the polygon = 360 º ÷ n. This angle is the angle of the triangle at the center of the polygon.
3. The sum of the interior angles in a triangle = 180º
4. The sum of the 2 base angles of each triangle must be 180 º - (360 º÷ n).
5. Each of the base angles of the triangle must measure (180º - (360 º÷ n)) ÷ 2.
6. Two adjacent base triangles from adjacent triangles form a vertex angle, so the vertex angle = 2* (180º - (360 º÷ n)) ÷ 2.

Two adjacent base angles = one vertex angle.

The vertex angle = 2 * (180º - (360 º÷ n) ÷ 2, which means you do not have to divide by 2. You can stop at step #6.

Now let's try a sample problem. Here is a site for constructing a pentagon. http://www.kjmaclean.com/Geometry/PentConstruct.html

Find the relationship between a pentagon's number of sides and its number of vertices and angles? Since a penta means 5, a pentagon has 5 sides, 5 vertices, and 5 angles. Now let's find the vertex angle.

1. Since a penta means 5, a pentagon has 5 sides, 5 vertices, and 5 angles.
2. 5 angles
3. 360º ÷ 5 = 72º
4. The angle of the triangle at the center of the polygon = 72º
5. 72 º + nº + nº = 180º
6. 180 º - (360 º÷ 5) = 108º
7. Each of the base angles of the triangle must measure 108º ÷ 2 = 54 º
8. The vertex angle = 2* 54 º = 108º
9. Notice that the answer to #6 and #8 are the same.

Answer 2

From small to greatest