Your turn
MathsKS3-GCSE
The Polygon Angle Brain Teaser
Every interior angle of a regular polygon measures 156°. How many sides does the polygon have?
Show the worked solution
What's the puzzle?
Every interior angle of a regular polygon measures 156°. How many sides does the polygon have?
What's the answer?
15 sides
How do you work it out?
- Each interior angle and its exterior angle add up to 180°, so each exterior angle is 180° - 156° = 24°.
- The exterior angles of any polygon add up to 360°.
- In a regular polygon all the exterior angles are equal, so the number of sides is 360 ÷ 24 = 15.
- The polygon has 15 sides. Check: (15 - 2) × 180° ÷ 15 = 156°.
Frequently asked questions
Why use exterior angles in the polygon angle puzzle?
The exterior angles of any polygon always add up to 360°, so in a regular polygon the number of sides is 360 divided by one exterior angle. That is quicker than rearranging the interior angle formula, which makes it the neatest route through the polygon angle puzzle.
Does the polygon angle puzzle work with the interior angle formula too?
Yes. The interior angles of an n-sided polygon add up to (n - 2) × 180°, so solving (n - 2) × 180 = 156n also gives the answer. LoveMaths Tutoring UK uses the puzzle to compare both methods.
Last updated: