Everyone learns the number one rule about triangles in high school: the angles of a triangle add up to 180 degrees.
But did you know that sometimes they don't? Triangles are much more interesting than you ever imagined. We're going to disprove this fact with one picture.
Keep in mind, however, that the above rule (namely, that the angles of a triangle add up to 180 degrees) is not false. In fact, it is true — however, it is only true in Euclidean geometry. That's the geometry that you learned in high school. In Euclidean geometry, the problems and shapes deal with flat surfaces.
But the world isn't flat. So we're going to take a look at non-Euclidean geometry, or the geometry that deals with 3D surfaces such as spheres or saddle shapes. That's where things like triangles start to get weird.
Take a look at the larger triangle on the left.
What you're seeing is a triangle that is drawn on the earth, a 3D sphere. You can see that the bottom two angles are both 90 degrees and that the top is 50 degrees. The sum of these angles is 230 degrees, not 180.
Why is this happening? Take a look at the two lines at the equator — they look parallel. However, in elliptic geometry there are no parallel lines because all lines eventually intersect. On a sphere, the natural analogue of a line on a flat plane is a great circle, like the equator or the longitude lines on a globe — they circle around the whole earth.
So when you extend these "lines" that are seemingly parallel to each other, they eventually have to intersect. And when this happens, naturally these lines form additional angles, such as the 50 degree one at the North Pole.
You're probably wondering about the second triangle on the right. As you can see, its angles add up to 180 degrees even though it is drawn over the curvature of the earth. Why is that? Well, this triangle is very small relative to the earth's surface (unlike the triangle to the left). As a result, it is less affected by the curvature of the earth.
Confusing? Not really. Imagine you drew a triangle with a piece of chalk on the sidewalk. It looks like you drew it over a flat surface, but the earth is still technically a curved surface. So your triangle — just like the one above — can seem more like a Euclidean geometry triangle because when you get really close to a curved surface, it looks flat.
In other words, the larger the triangle is on a spherical shape, the more of the curvature of the earth it will cover. And then everything you thought you knew about basic triangles starts to change.
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