Generalizing and Finding a Pattern
Let's recap what we have discovered so far
| Number of Sides of Polygon | Number of Triangles in Polygon | Sum of the Interior Angles |
| 4 | 2 | 180 |
| 5 | 3 | 180 |
| 6 | 4 | 180 |
| 7 | 5 | 180 |
| 8 | 6 | 180 |
| n | ? | ? |
Do you notice a pattern?
How does the number of sides of a polygon compare to the number of triangles that can be drawn in that polygon? (How does the number in the first column compare to the number in the second column?)
If the number of sides of a polygon is , then how many triangles can be drawn in that polygon? Write your answer in terms of .
What number are we always multiplying the amount of triangles in a polygon by to get the sum of the interior angles in a polygon? (hint: what number always appears in the third column of the table?)
If the number of sides of a polygon is , then what is the sum of the interior angles of that polygon? Write your answer in terms of . (Hint: use your answer from question 2)
And that's it!
You have just found the formula to find the sum of the interior angles in any polygon that has sides! Again, the formula for the sum of the interior angles of a polygon with sides is