# Using a pen to sum interior angles

What follows is a brief description of how to find the sum of interior angles of a simple convex polygon, since I haven't been able to find a YouTube video on this demonstration. Instruct the students to pay attention to the rotation of the pen. Have them note whenever the pen has rotated 180 degrees or more. Start with a triangle, such that one of its sides is horizontal. Place a pen, needle, or otherwise, along the base of the triangle. Slide the pen to the right until one of its ends is at one of the vertices of the triangle. Now rotate the pen about this vertex, clockwise, until the pen is once more aligned with a side of the triangle. Now slide the pen up and to the left, along the edge of the triangle, until it hits another vertex. Rotate clockwise about this new vertex, just as before, until the pen is aligned with the third side of the triangle. Carry out this procedure one last time, and the pen has returned to its initial position, but now the pen is backwards. And since the pen did not rotate more than 180° degrees throughout the process, it must have rotated 180° degrees, and thus, the sum of interior angles of the triangle must also be 180° degrees.

Now, an interesting question for your students: what can we infer about the sum of a quadrilateral's angles, from this process?" Maybe some students will realize that the sum must be a multiple of 180°, as the pen must return to its original position. Questioning them further, about the sum of interior angles for any simple convex polygon, they might realize this too must be some multiple of 180°.

As an aside, this method can also be used to find the sum of angles in the tips of a star. In this case, instead of sliding the pen along the boundary, you would slide the pen along the lines in which you would typically draw a star.

What other shapes might be interesting to measure in this way, if any?