My good friend who runs the wonderful blog In The Margins gave me this problem quite some time ago, and I honestly enjoyed completing it quite a bit. Thought I’d go ahead and share it with the world!

Problem:

“Find the area of a triangle on a sphere without using Calculus, given just its angles .”

I recommend pausing here and trying this yourself first, it’s quite fun!

Solution:

First, make sure you know what a great circle is. It’s the intersection of a plane (“paper”) and a sphere (“orange?”) where the plane goes through the center of the sphere. Look at the image to the right for an example (for instance, the Earth’s equator is a great circle in a sense!)

Begin by first drawing a triangle on a sphere with arbitrary angles . Afterwards, extend the sides of the triangle to form great circles. In doing this, you should notice that the same triangle appears on the back of the sphere.

Now denote the side between and as 1, and as 2, and and as 3. Now only looking at side 1, shade in the hemisphere not containing the triangle, and the repeat the process with the other two sides.

Let’s denote the following:

Set of Shaded Points from The Hemisphere of Side

Set of Shaded Points from The Hemisphere of Side

Set of Shaded Points from The Hemisphere of Side

Note then, that this follows:



(where is the area of the triangle)



(this, along with the following, are called antipodal digons)





Looks like we’re in a good situation to employ the Inclusion-Exclusion Principle!

Inclusion-Exclusion Principle for 3 Finite Sets:

Let , , and be three finite sets. Then



Which means we must have







Thus we have shown the following:

Proposition:

The area of a triangle on a sphere of radius with angles is

And there we have it! There’s our area. Oh! And also!

Corollary:

The sum of the interior angles of a triangle on a sphere must exceed .

Isn’t that so cool? Being used all your life to the sum of the interior angles of a triangle being precisely 180 degrees — how do you feel seeing that now it’s mandatory it exceed 180 degrees?