Friday, January 06, 2012

Polygon Angles

I've tried various ways over the years of presenting the following 2 facts:
* (n-2)*180 is the sum of the interior angles of a polygon.
* 360 is the sum of the exterior angles of any polygon.

Yes, the students remember ultimately, but as I was looking over my old files this year to do it again, things just didn't sit right ... or I was in the mood for a change ... or I'm a different person now, and I want to approach it differently.

I hit upon the following method, and I think I like it the best (ask me what I think next time). Here are things I like about it: it's not talk, talk, talk. It's not, "let's do it all theoretically and then POOF here's the formula." It's not, "let's all explore a different polygon by drawing and measuring and seeing what happens and there'll be human error, but we'll hand wave that away." It's pretty efficient. It's different from what I usually do, so it'll stick in MY students' minds (hopefully).

I had everyone get a piece of colored paper and fold it in 4ths and cut the 4 separate rectangles. Then on one rectangle, I had them draw a heptagon (we drew a light circle, put 7 dots on it, and connected the dots). We took a colored marker to the border of the heptagon and made sure that some color was on the interior. We drew diagonals from one vertex. We cut out the heptagon. Then we cut the triangles out.

Then I said, so what we want to explore is if I had asked you to measure all the interior angles and add them up what would you get. So now look at what we have done, and see if you can figure out the answer and then ALSO figure out why I asked you to do these particular tasks.

Then I waited and eavesdropped. Then I asked a kid to explain. Then I had them tape this into their composition books to look something like this:

Then (after some other polygon stuff), on another piece of the colored paper, I had them draw an octagon but extend the sides to show the exterior angles. Before I had them do the following thing, I had them take thumb votes on the scenario, "as the number of sides increases, what do you think happens to the sum of the exterior angles as shown: thumbs up for increases, down for decreases, sideways for stays the same, or open palm for it depends on other factors." They thought and voted. Then we did....

Then I had them cut out the interior octagon and leave the border. We reviewed what the sum of the interior angles was (6 * 180). Then we built up what the sum of all the "lines" angles were (8 * 180). Then we thought and discovered and here's what they taped in their notebooks:

Then as always I ran out of time and was rushed and couldn't do any practice problems, so my resolution of stopping class 1 minute before the bell so they could pack up was not put into place today!


  1. I've always liked the "turtle graphics" way of showing that the exterior angles add to 360. The Scratch programming language is a good way to do that.

    1. How would you do that using the Scratch program?

  2. That triangle slices is very slick. If that isn't sticky enough to last I don't know what is. Do you help them see that there is always going to be n-2 triangles?

  3. Hmmm, I haven't heard of "turtle graphics". I'll Google it. Thanks for the tip.

    Fall Garlic: yup, we talk through and I have them envision another polygon and they think and we conclude that there are always 2 less triangles.

  4. Once the kids cut out the three angle tips of a triangle and put them together, they are okay that this proves the 180 sum. Then they draw in triangles in polygons with 4 sides and higher and again are satisfied that the number of triangles is always two less than n-sided polygons. I like your idea of cutting out the triangles, definitely more tactile.

  5. Scratch (which uses the 'turtle graphics') is free at And there's a huge community of users sharing ideas.

    I like your way of showing the exterior angles add up to 360 degrees, and I like the turtle way too. They're so different, I think that might intrigue some kids.

  6. Anonymous2:44 PM

    Is the "turtle graphics" way a similar idea to this:

    I'm walking along the raised edge of a garden bed, balancing as if on a gymnastics beam. When I get to a corner, I will NOT stumble off the end and have my friends point and laugh. Instead I will turn through (the exterior angle) and continue to walk along the next side. No matter how many sides the garden has, by the time I get back to the starting point I have spun through 360.

    I use this with my kids, and it really does click - especially when I act out balancing (arms out, wobble wobble) and stumbling off the end 'cause I wasn't expecting the last step to be a doozy!