My younger son and I have just started talking about some basic geometry. He has so many neat ideas which make talking about geometry with him super fun.

Today we were returning to a topic from a few days ago – angles in polygons. The specific topic for today was exterior angles. For the first part of our talk today we reviewed what happens with interior and exterior angles in a triangle:

Next up was quadrilaterals – what happens with exterior angles here? The fact that the interior and exterior angles here have the same sum was a little confusing to him – his initial reaction was that the two sums should never be the same.

After expressing some skepticism, he proceeds to calculate the sum of the exterior angles a different way and finds that the sum is indeed 360 degrees. Having done calculation two different ways, he now believes the sum is 360 degrees and even begins to wonder if you get 360 degrees as the sum of the exterior angles for any polygon.

Since he was looking to generalize, we moved on to studying pentagons. He’s getting comfortable now with the idea of chopping polygons up into triangles, so he sees that the sum of the interior angles is 540 degrees. He then does a similar calculation to what he did in the last two videos to compute that the sum of the exterior angles is 360 degrees.

Now he really believes that the sum of the exterior angles of any polygon will be 360 degrees. Fun 🙂

One thing that made me really happy about this short project was his skepticism about the sums for the quadrilateral. The answer didn’t seem right to him, and he looked for an alternate way to find the answer. When he found the same answer the second time around not only did he have the confidence to believe it, he began to think that there might be something more general happening and wanted to check the next step. Love the curiosity of kids!