Interior angle sum of polygons: a general formula

Activity 1: Creating regular polygons with LOGO (Turtle) geometry

For this activity, click on LOGO (Turtle) geometry to open this free online applet in a new window. (Or alternatively download to your computer StarLogo turtle geometry from the Massachusetts Institute of Technology (MIT) for free by clicking on the link.) Try to resize the windows a little so you have them more or less alongside each other.

With this nice applet you can program a little turtle to move FORWARD with the command 'fd', and turn it LEFT or RIGHT through any angle in degrees you choose with the respective commands 'lt anglesize' or 'rt anglesize', to DRAW an endless variety of geometric figures.

Instructions:
1) In the active window of the open applet, first click the 'clear' button on the right column to Clear Program.

2) Then in left column/margin below PROGRAM type the following commands to move the turtle FORWARD by 200 steps and TURNING it RIGHT through 144 degrees, and repeating it another 4 times. Can you guess what type of polygon will be formed?
fd 200
rt 144
fd 200
rt 144
fd 200
rt 144
fd 200
rt 144
fd 200
rt 144

3) Now click on the 'run' button to Run Program. What type of polygon is formed? Is that what you expected? Can you figure out the sum of its interior angles?

4) To draw a new figure, click on the 'clear' button to clear the screen. Try drawing an equilateral triangle, a square, a regular convex pentagon, etc. Or try repeating these procedures a number of times and see what figures you get: forward 150 right 135; forward 150 right 160; forward 150 right 80, etc. Can you figure out the sum of the interior angles of each of these figures?

5) What will the sum of the interior angles be if the polygon in 3), or any of the others, are irregular, and do not have equal sides and angles? Will their interior angle sum change or be the same? Continue below or investigate further using suitable dynamic geometry software!

Activity 2: Investigating a general formula for the sum of the interior angles of polygons

1a) You may have earlier learnt the formula S = 180(n-2) by which to determine the interior angle sum of a polygon in degrees, but this formula is only valid for simple convex and concave polygons, and NOT valid for a star pentagon like the one shown below.

star pentagon

1b) From Activity 1, no. 3, make a conjecture regarding the interior angle sum of a general star pentagon? Can you explain (prove) it?

1c) Click on the link below to check the interior angle sum for a star pentagon like ABCDE.

NOTE FOR EACH OF 4 LINKS BELOW: If a security pop-up menu appears in your browser, please choose RUN/ALLOW to let the GeoGebra applet run properly. It is completely safe & can be trusted. If you have the very latest Java on your PC or Apple Mac, and experience problems with the Geogebra applets loading, please go here for additional information on Java settings that should resolve the issue.

Star pentagon

1d) Drag vertices A, B, C, D or E. What do you notice about the Angle Sum of the star pentagon in the left column? Can you explain (prove) it?

1e) Though there are several different ways of proving the interior angle sum of a star pentagon like this, try reasoning from rotating a pencil by the exterior angle at each vertex in order, and counting the total number of full revolutions it undergoes, moving completely around the figure until facing in the same starting direction.

Challenge: Can you find a general formula by which to easily find the interior angle sum of the star pentagon above, as well as the following polygons?

More crossed polygons

2) Click on the link below to check the interior angle sum for a crossed septagon like ABCDEFG. Then drag the septagon into other shapes and see if you can correctly predict or explain the shown interior angle sum.

Crossed septagon

3) Click on the link below to check the interior angle sum of a crossed octagon like HIJKLMNO. Then drag the octagon into other shapes and see if you can correctly predict or explain the shown interior angle sum.

Crossed octagon

4) Click on the link below to check the interior angle sum for a crossed quadrilateral like PQRS. Then drag the quadrilateral into the shape of a convex or concave quadrilateral, and then back again into a crossed shape. Can you explain what you observe?

Crossed quadrilateral

Note: My book Rethinking Proof has a learning activity with a worksheet and associated sketch in relation to the defining of 'interior angles' in terms of the concept of 'directed angles' so as to extend it in a consistent way to the interior angle sum of a crossed quadrilateral, and crossed polygons in general.

Also read my article Stars: A second look for a short proof of a general formula, and for more examples, see Problem 15 on pp. 49-50 and the Solutions on pp. 135-136 of my book Some Adventures in Euclidean Geometry.

Michael de Villiers, 25 January 2011, created with GeoGebra