Did I hear someone say “I already knew that!”.  Judging from the names of triangles we learnt in (Primary School) elementary school,below

• scalene
• isosceles
• equilateral
• acute
• obtuse
• right(-angled)

this seems like an question in “Are you smarter that a 5th Grader?”  But I assure you this is not.  Dr. Yoichi Maeda published
Seven Types of Random Spherical Triangle on the journal of Computational Geometry and Graph Theory in 2007, but all he was interested in were acute or obtuse angles.  But isn’t there just two (or three) kinds of angles?  Either acute or obtuse?  Or on a very rare situation, right-angled?

Of all the triangles we know in plane geometry, the familiar right-angled one is the one that gives us the famous Pythagoras Theorem with $a^2+b^2 = c^2$.  But probabilists said those were very rare and constitute only a set of measure ZERO.  And Yoichi is only interested in triangles on $S^n$ like for example, $S^2$, which is a sphere like the surface of the Earth.  Now that is interesting, as navigators (air or sea) they like to draw “straight lines” between destinations and form triangles.  I am sure they are interested in the seven types of triangles Yoichi is referring to.  And as Earthings we should be too.

It turns out that $a^2+b^2 = c^2$ does not hold for triangles on a surface like the Earth!  In fact, angles don’t even add up to a hundred and eighty degrees  — 180o.  Take for example the following trtiangle ABC

A and B are two different points located on the equator.  Now make make A and B go due north and they will not meet each other until they both reach the north pole, which we will call C.

The above triangle will have two right angles (one at A and one at B).  And adding the third angle at C, we will inevitably end up with a triangle whose sum-of-angles is more than 180o.  For that reason, triangles on a sphere could be

• acute-acute-acute
• acute-acute-obtuse
• acute-obtuse-obtuse
• obtuse-obtuse-obstuse

if we agree to omit those triangles where on of its angle is a right angle because they are of measure ZERO anyway.  So that is 4 different kinds of triangles on the surface of the sphere,  where are the other 3 kinds?  It turns out that even the length of the triangles matter in analysis on the surface of the sphere.  Let’s say the radius of the earth is R (that would be 6371 km), then we can a side of a triangle obtuse if it is larger than $6371 \times \frac{\pi}{2}$ (or we can say it is OBTUSE if it is more than ‘radian measure of’ 90oafter resizing it by dividing by R).  Since there are three sides to a triangle we can say triangles on a sphere could be

• ACUTE-ACUTE-ACUTE
• ACUTE-ACUTE-OBTUSE
• ACUTE-OBTUSE-OBTUSE
• OBTUSE-OBTUSE-OBTUSE

Then together, we would have too many kinds of triangles.  But looking at it more closely with more reasoning, he was able to reduce the number of categories to only seven kinds because for instance and OBTUSE-OBTUSE-OBTUSE is naturally also an obtuse-obtuse-obstuse.

what kind of triangle