# Triangles in planes and spheres

Today our fun Family Math project was about geometry.  We did a little playing around with triangles in the plane and triangles on the sphere.   A more advanced version of this discussion would probably include some mention of Euclid’s 5th postulate.

Our first topic of discussion was parallel lines on a plane.  What does it mean to be parallel?  My youngest son sees parallel lines as lines that do not intersect and my oldest wants to define parallel in terms of the slope of the line.

After talking about parallel lines for a bit, we went on to talk about parallel lines and angles:

Next we go on to talk about triangles.  The point of this discussion is to see that the angles in a triangle can be rearranged to make the same angle as a straight line.   The main idea here is just the idea that we discussed in the last video:

Now we move on to some fun ideas about triangles.   Just using some of the  basic facts about angles that we talked about in the last movie + the Pythagorean theorem, we find the area of a equilateral triangle, and also some simple properties of an equilateral right triangle:

Finally the punch line – what happens if we try to extend some of these geometric ideas beyond the plane?  The easiest example to show is a sphere, and I illustrate a triangle with three right angles by drawing the picture on a softball.  I love that my youngest son’s reaction was that this triangle was impossible.  Ha, not impossible, you are looking at it right now!!

Feels like there are a lot of different directions to go introducing basic geometric ideas to young kids.  One unexplored idea here is to show a surface where a triangle’s angles add up to less than 180 degrees.   Maybe there’s a 3D printing / basic geometry project in the near future!