Interior Angles with Triangles (and Circles) #MTBoS30 9 of 30

The foundation:

The are only a few givens in geometry, building on those we derive many other patterns theorems.  One theorem I often prefer to focus on visually is the sum of the interior angles of a convex polygon.  There’s different ways to approach the process, and most of them refer to creating a fan of triangles inside the polygon:

Early on in most geometry studies, we learn that the sum of the interior angles of a triangle is equal to 180, building off of this we can use the above animation to calculate the sum of the convex hexagon: (180)(4)=720.

The Detour

There’s often the student who wants to divide the hexagon up differently.  The student wants to draw lines criss crossing all over the shape.

 

The Brakes:

Too often I would have totally passed up this opportunity and said, “That’s not how we do it, so it can’t work.”

Taking The Long Way isn’t an Error:

Thankfully, this year we saw something different.  There were still triangles inside, and with a few more lines added, the picture had only triangles inside.

Once we have all triangles, we can just count those and multiply by 180.

 

(16)(180)=2880.  That can’t be right, can it?

From here we get a better definition for what is an interior angle of a hexagon.  It’s an angle inside the hexagon, but still attached to the edge of the shape.  Somehow we need to get rid of the angles that are inside the hexagon but not touching the edge of the shape, like these red circles:

 

There are 6 of these circle sets of angles, each of them having a value of 360 degrees.  We just need to discount those (aka subtract).

2880-6(360)=2880-2160=720.

This kind of process also builds a more flexible understanding of how to decompose a geometric shape in multiple ways.  I encourage all math teachers out there to try this next time a student chooses to slice it up however he/she wants.

By the way, thanks to @justinaion  for inspiring this with his post earlier today.