How Many Degrees Are In A Triangle?

In other words, what is the sum of the interior angles of a triangle?

After thinking way back to geometry (further for some than for others), we can recall (or look up) that there are 180 degrees in any triangle. This idea is the basis of many proofs high school students attempt in their mathematical careers, so many are familiar with it.

But is it a lie?

If you're most people (and most of you are), then you probably deal almost exclusively with Euclidean geometry. You can look it up on Wikipedia for a thorough description, but for our purposes it basically means that your shapes are drawn on flat, smooth surfaces.

So when we look at triangles in Euclidean geometry, all of the interior angles of a triangle add up to 180 degrees, just as we were taught.

But what if the triangle is drawn on a surface that isn't flat? And where on Earth would I find such a triangle?

There's the beauty. You find that kind of triangle everywhere on Earth.

Take a look at this globe.

Let's imagine drawing a triangle. Side A runs along the equator. Side B runs from the North Pole to the equator (at 20E, or 20 degrees to the right). Side C runs from the North Pole to the equator (at 40E, or 40 degrees to the right).

Now that we know our three sides, let's get the three angles.

The angle between Side B and Side C, Angle A, is 20 degrees (the difference between 40 and 20 is 20).

Now for Angle B and Angle C. Side B and Side C are lines of longitude and Side A is a line of latitude. Lines of latitude and lines of longitude are perpendicular (another geometry word). This means they make right angles (or angles of 90 degrees).

Because of this, we know the angle between Side A and Side B, Angle C, is 90 degrees.

We also know the angle between Side A and Side C, Angle B, is 90 degrees.

So to recap:

Angle A = 20 degrees

Angle B = 90 degrees

Angle C = 90 degrees

Add them up and we get 200 degrees! These triangles exist in what is referred to as elliptical geometry, not Euclidean.

A slightly more complicated example shows a triangle with less than 180 degrees in what is known as hyperbolic geometry (you can read more on that on your own).

So there we have it. Sometimes a triangle has 180 degrees, but sometimes it doesn't. And you don't have to look beyond Earth to find one that doesn't.

Anyway, I hope you got something out of this. If not, well, there's always next time.

If you have any questions, or if anything isn't clear, just let me know in the comments.

I stumbled across this in the book The Big Questions: Mathematics. (ISBN: 978-1-4351-1131-8).

The globe image was found on http://kaffee.50webs.com/Science/index.html.