Geometry & Measures
Grade 4-6Angles in Polygons
Angles in polygons build directly on the angle sum of a triangle, extending the idea to shapes with any number of sides, and examiners often combine this with algebra by hiding an unknown angle inside an equation. This lesson covers the sum of interior angles of any polygon, the interior and exterior angles of regular polygons, and using these facts to find a missing angle or number of sides.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Sum of interior angles
Any polygon can be split into triangles by drawing diagonals from one vertex, and since each triangle contributes 180°, the sum of the interior angles of a polygon with n sides is (n - 2) × 180°.
A polygon with 9 sides has an interior angle sum of (9 - 2) × 180 = 1260°. If several interior angles of a polygon are known and one is unknown, subtract the known angles from this total to find the missing one.
Interior and exterior angles of regular polygons
In a regular polygon, all interior angles are equal, so each interior angle is the total interior angle sum divided by the number of sides: (n - 2) × 180 ÷ n. Each exterior angle is 180° minus the interior angle, and the exterior angles of any polygon always sum to 360°, so each exterior angle of a regular polygon is 360 ÷ n.
A regular nonagon (9 sides) has each interior angle equal to (9 - 2) × 180 ÷ 9 = 140°, so each exterior angle is 180 - 140 = 40°.
Finding the number of sides
If the exterior angle of a regular polygon is known, the number of sides is 360 divided by the exterior angle. If only the interior angle is given, first subtract it from 180° to find the exterior angle.
A regular polygon has an interior angle of 140°, so its exterior angle is 180 - 140 = 40°, giving 360 ÷ 40 = 9 sides.
Combining polygon angles with algebra
Some questions give interior or exterior angles as algebraic expressions rather than numbers, requiring an equation to be formed and solved using the interior angle sum or the fact that exterior angles sum to 360°.
The exterior angles of a pentagon are 2x, 3x, x + 10, 4x and x + 20. Since exterior angles sum to 360°, 11x + 30 = 360, so x = 30.
Worked Examples
Three exam-style questions, fully solved.
Find the sum of the interior angles of a polygon with 9 sides.
Easy- 1.Use the interior angle sum formula: (n - 2) × 180 with n = 9
Answer: 1260°
A regular polygon has an interior angle of 140°. Find its exterior angle, and hence the number of sides of the polygon.
Medium- 1.Find the exterior angle: 180 - 140
- 2.Divide 360° by the exterior angle to find the number of sides: 360 ÷ 40
Answer: 9 sides
The interior angle of a regular polygon is 8 times its exterior angle. Find the number of sides of the polygon.
Hard- 1.Use the fact that interior and exterior angles sum to 180°: 8 × exterior + exterior = 180
- 2.Simplify and solve for the exterior angle: 9 × exterior = 180, so exterior = 20°
- 3.Divide 360° by the exterior angle: 360 ÷ 20
Answer: 18 sides
Avoid These
The most common mistakes students make.
Using n instead of (n - 2) in the interior angle sum formula, forgetting to subtract 2 for the two triangles that cannot be formed at the starting vertex.
Confusing the interior and exterior angle formulae, for example dividing 360° by the number of sides to find the interior angle instead of the exterior angle.
Forgetting that exterior angles of any polygon, regular or irregular, always sum to 360°, and instead trying to use the interior angle sum formula for exterior angles.
When an interior angle is given, forgetting to subtract it from 180° first before dividing 360° by the result to find the number of sides.
Making an arithmetic slip when substituting into (n - 2) × 180 ÷ n, especially forgetting the order of operations when a calculator is not allowed.
FAQ
Questions parents and students ask.
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