Problem
Splitting from one vertex into 4 triangles gives 6 interior angles summing to 4 x 180 = 720 degrees, so each equals 720 / 6 = 120 degrees.
Cutting a polygon into triangles is exactly the Grade 4 way to find angle sums; each triangle is 180 degrees.
4.G.A.2Solve An Easier Related ProblemOn the straight line, interior + exterior = 180, so each exterior angle is 180 - 120 = 60 degrees.
On a straight line the two angles must fill 180 degrees, so subtraction gives the leftover exterior angle.
4.MD.C.7Draw A DiagramEach exterior angle of the regular hexagon measures 60 degrees.
At each vertex the side and the extension of the next side lie along one straight line, so the interior angle and the exterior angle together fill 180 degrees.
That interior angle measures 120 degrees, which is the part of the 180 that the exterior angle does not take up.
The six interior angles of the hexagon add up to 720 degrees.
Diagonals drawn from one vertex cut the hexagon into 4 triangles, and the corner angles of those triangles fit together exactly to form the six interior angles with no gaps or overlaps.
The three corner angles of each triangle add up to 180 degrees.
The triangle corners cover the interior angles completely without any left out or counted twice, so the pieces add back to the whole.
Four triangles that each hold 180 degrees give 4 equal groups of 180, which is 720.
Because the hexagon is regular all six interior angles are equal, so each one is the 720 shared out into 6 equal parts, and splitting equally like this reverses multiplying.
So the exterior angle is what is left when the interior 120 is taken away from the straight 180, and subtraction finds that leftover: 60.
All six exterior angles are equal to 60 degrees, so the sum is 6 x 60 = 360 degrees.
Adding six equal angles is just repeated addition (multiplication), a skill young learners already have.
4.MD.C.7Identify SubproblemsWalking once around any polygon turns you a full circle, so the outside angles always add up to 360 degrees!