http://www.ck12.org Chapter 6. Polygons and Quadrilaterals
Guidance
Recall that interior angles are the angles inside a closed figure with straight sides. As you can see in the images
below, a polygon has the same number of interior angles as it does sides.
A diagonal connects two non-adjacent vertices of a convex polygon. Also, recall that the sum of the angles in a
triangle is 180◦. What about other polygons?
Investigation: Polygon Sum Formula
Tools Needed: paper, pencil, ruler, colored pencils (optional)
- Draw a quadrilateral, pentagon, and hexagon.
- Cut each polygon into triangles by drawing all the diagonals from one vertex. Count the number of triangles.
Make sure none of the triangles overlap.
- Make a table with the information below.
TABLE6.1:
Name of Polygon Number of Sides Number of 4 sfrom
one vertex(Column 3)×(◦in
a 4 )Total Number of
Degrees
Quadrilateral 4 2 2 × 180 ◦ 360 ◦
Pentagon 5 3 3 × 180 ◦ 540 ◦
Hexagon 6 4 4 × 180 ◦ 720 ◦- Do you see a pattern? Notice that the total number of degrees goes up by 180◦. So, if the number sides isn, then
the number of triangles from one vertex isn−2. Therefore, the formula would be(n− 2 )× 180 ◦.
Polygon Sum Formula:For anyn−gon, the sum of the interior angles is(n− 2 )× 180 ◦.