KA-0170Angles on lines, in triangles, in polygons
3 points, difficulty 1 of 3Level 11–12about 75sWhat is the size of one interior angle of a regular twelve-sided polygon?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
The exterior angles of any polygon always add to 360 degrees.
2The strategy
Each exterior angle of a regular 12-gon is 360 / 12 = 30 degrees.
3The full solution
Interior and exterior angles sit on a straight line, so the interior angle is 180 - 30 = 150 degrees.
Solution
The reliable way
The exterior angles of any convex polygon sum to 360 degrees, so in a regular twelve-sided figure each exterior angle is 360 / 12 = 30 degrees. An interior angle and its exterior angle lie on a straight line, so the interior angle is 180 - 30 = 150 degrees. Cross-check with the other formula: the interior angles total (12 - 2) x 180 = 1800 degrees, and 1800 / 12 = 150. The transferable idea: the exterior route is shorter and almost always safer.
The elegant way
Exterior angles share 360 degrees twelve ways, giving 30 each, so the interior angle is 150.
Why this is on the test: Answering with the exterior angle, or with the total rather than one angle, are the two errors this question is built to catch.