How it works
A polygon with n sides can be cut into n − 2 triangles from one vertex, so
Sum of interior angles = (n − 2) × 180°
In a regular polygon all angles are equal, so each interior angle is (n − 2) × 180° ÷ n. The exterior angles of any convex polygon add up to 360°, so each exterior angle of a regular polygon is 360° ÷ n. At every vertex, interior + exterior = 180°.
Exterior angles are usually the fastest route: a regular polygon whose interior angle is 150° has exterior angle 30°, so it has 360 ÷ 30 = 12 sides. The number of diagonals is n(n − 3) ÷ 2, because each vertex connects to all but itself and its two neighbours, and each diagonal is counted twice.
Polygon Angles practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
Find the measure of each exterior angle of a regular decagon in degrees.
Show answer and solution
Answer: 36°
The exterior angles add to 360°, and a decagon has 10 of them: 360 ÷ 10 = 36°.
The sum of the interior angles of a polygon is 1260°. Find its number of sides.
Show answer and solution
Answer: 9
(n − 2) × 180 = 1260 gives n − 2 = 7, so n = 9.
A regular polygon has each interior angle 3 times its exterior angle. Find its number of sides.
Show answer and solution
Answer: 8
Let the exterior angle be e. Interior + exterior = 180, so 3e + e = 180 and e = 45°. The number of sides is 360 ÷ 45 = 8.
A polygon has 35 diagonals. Find its number of sides.
Show answer and solution
Answer: 10
n(n − 3) ÷ 2 = 35, so n(n − 3) = 70. Since 10 × 7 = 70, n = 10.
Keep going
Related topics
- Triangle inequality
- Special right triangles: 45-45-90 and 30-60-90
- Arc length and sector area
- Similar triangles and scale factor
Related practice sets in the app
All Geometry topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice