Sum of Polygon Angle Measure Investigation
We have already discussed the Triangle Angle-Sum Theorem, which tells us the measure of all angles included in a triangle add up to 180 degrees. Now, let's take a look at some other polygons and see if we can discover a rule which will tell us how many degrees are in an n-sided figure! For each figure below, do the following things:
- Add up the total degrees of the interior angles of the figure as you see it now.
- Click and drag one of the vertices to a new location. Notice that as you move the corner, the angle measures of the figure change with you. Now, add up the interior angle measures again to get a new total.
- move two more vertices of the same figure to create a third shape with the same number of sides. Again, calculate the total degrees of the interior angles of the polygon.
- What do you notice for each of these three examples? Were the angle measure totals the same each time or different?
Once you've completed this task for each figure, answer the following questions:
- Multiply the numbers 1 through 5 by 180. From here on out, we are going to call the numbers 1-5 our "mystery" numbers. Record these mystery numbers in a table with the original number in the left column and the result after multiplying by 180 in the right column
- For each of the figures you examined above, find the total number of degrees you calculated. Which row in your table does this value correspond with (that is, which original number is paired with that angle measure in your table?)
- Now for each figure, note how many sides it has. What relationship do you see between the number of sides of the figure and the mystery number that gave its angle measure?
- Suppose we had a figure with 6 sides. SHOW OR EXPLAIN how you would find the sum of the interior angles of this figure.
- Suppose we had a figure with n number of sides. SHOW OR EXPLAIN how you would find the sum of the interior angles of this figure.