Good morning! I’m teaching Geometry this semester, so triangles are on my mind. And here are some facts about triangles that, even though I know them, still blow me away.

I was taught that the angles in a triangle add up to 180 degrees. This is true, at least in Euclidean Geometry, which is geometry done on a flat surface like a plane or piece of paper..(That link was to an illustration using folded paper but there are many proofs too.)

But what if you’re not on a flat surface? What if you’re on a sphere? Then things get weird. “Lines” on a sphere are defined by the shortest path between two points (formally called geodesics), and it turns out that if you draw a line between two points and keep going, it will cut the sphere in half, like the equator of a globe, or a line of longitude. And if you put three of those together you get a triangle…but the angles don’t add to 180 degrees. In fact, you can have have a triangle with not one but two right angles, as shown below:

You can even have a triangle with three right angles! Or three obtuse angles! And overall, there isn’t a fixed amount that the angles add up to: it can be anything from just over 180 degrees all the way up to just under 540 degrees (which would be a triangle with three really big angles, covering a significant amount of the sphere). So weird.

That’s not the only thing that’s weird. If, instead of being on a sphere (which bows out), you are on a hyperbolic paraboloid, which bows in and looks like a saddle, then everything is opposite and the angles of a triangle are smaller than they would be on a flat surface. You can still have triangles with one right angle or one obtuse angle, but the other two angles will be a bit smaller, and some triangles will just have three very small angles. As in spherical geometry, the angles of a triangle don’t add up to any one fixed amount, but can be any positive number less than 180 degrees.

If you like exploring, the (free) program Geogebra [that mimics drawing with a straightedge and compass] has tools that let you draw in spherical geometry and hyperbolic geometry. Enjoy!