12 point circle
Use the line segment tool or the polygon tool with the points on the circumference of the circle to draw an equilateral triangle.
How many different equilateral triangles can be drawn?
What are the numbers of the three points on each equilateral triangle?
(use the arrow tool to hit the reset button).
Similarly, how many squares? What are the numbers of the points?
What about hexagons? How many? What are the numbers of the points?
Any other regular polygons?
9 point circle
What different regular polygons can be drawn using a 9 point circle? How many of each?
15 point circle
What different regular polygons can be drawn using the 15 point circle? How many of each?
24 point circle
What different regular polygons can be drawn using the 24 point circle? How many of each?
Star Diagrams Project
Let’s return to the 12 point circle. Starting at zero, join every fifth point. So, 0 to 5 to 10 … go past zero to 3 …then to 8, to 1 etc. Count the number of times you go around the circle until you arrive back where you started. The result should be a nice 12 point star.
Here’s an organised way to record some results: Star Drawings Record Sheet. On each page there is a diagram for just one drawing, however the results table is for all possible increments. Use the applet above to join every 1 point, then to join every 2nd point, then to join every 3rd point etc. Record a description of each resulting diagram in the table. Choose one diagram to draw onto your record sheet.
Now let’s go to the 9 point circle. Start at zero and join every 1st, then 2nd, then 3rd and so on. Notice the increments that result in a regular polygon, and those that result in a star.
The same for the 15 point circle. By now you probably know which increments will result in a regular polygon, and which ones result in a star. But why do some stars use all points and some stars only a few points?
Before you investigate the 24 point circle, make some predictions. Decide which increments will result in regular polygons and state the kind of regular polygon. Decide which increments will yield a 24 point star, and which will yield some other point star. Try to figure out in advance how many points each star will have!
Test out your hypotheses on the applet, and write up your conclusions with any new insight you have.
And trigonometry – if you’ve studied sines, tans and cosines, you’ll be able to figure out how long these chords are too.