by tommywallach Thu Jan 01, 2015 6:26 pm
Hey Kg,
In a regular figure like that, you can assume that dividing up into pieces will create equal sized shapes, because any asymmetries would be illogical (because the resultant figure wouldn't be the same on all sides, and thus would no longer be "Regular"). That said, I totally agree with you that the explanation is a little light. I'll look into fleshing it out.
For now, if you really want to PROVE it, you'd have to put a length on all the sides (say 6, as an example). Then, start solving for the areas of the resultant triangles (you can do that because these are 60/60/60 triangles, which means they can be split into two 30/60/90 triangles, and if you know one side of such a triangle, you can get the base and the height, so you can solve for the area). By the way, you shouldn't need to do all that work, because of what I said in the first paragraph. But that's how you'd "prove" it.
Hope that helps!
-t