How many degrees in a shape?

Even if it's been ages since high school geometry, I imagine that most people still remember how many degrees there are in a triangle.  (Just in case you forgot--it's 180 degrees.)


Remember?  There you go!

Now, how many degrees are there in square?--That's 360 degrees.

Now, what about a pentagon?  I don't know if I ever even learned that one in high school!  It's easy, though, to determine how many degrees there are in any convex polygon.  One method is to use a process called triangulation, which is essentially drawing triangles inside the polygon, all originating from one point. If you draw a line segment from one point to every other point in the triangle (except the two points closest because there is already a line there), you will draw triangles.  Since every triangle's interior angles sum to 180 degrees, you can count the number of triangles you have just drawn in the polygon and then multiply that number by 180 degrees to get the total number of degrees in the polygon. Check out this method from Khan academy

Back to that first assumption.  How do we know there are 180 degrees in a triangle?  Is this something that the math teachers know and that we everyday non-mathy people just can't understand?  The answer is a resounding NO! There's an easy way to prove that there are 180 degrees in a triangle, just click on the Geogebra image below to play with an interactive geogebra tube and prove it to yourself!

"Triangle.Equlilateral.png" by en:user:herbee is licensed by CC under the GFDL

"GeoGebra" by gengebra.com