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.
What you need to know
- The exterior angles of any polygon add to 360 degrees.
- One exterior angle of a regular polygon is 360 divided by the number of sides.
- The interior angles add to (n minus 2) times 180 degrees.
- At each vertex, the interior and exterior angle add to 180 degrees.
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
Practice
6 questions, marked instantly
Type an answer and check it. The worked solution appears once you have had a go. No login needed, and your progress saves in this browser.
Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Work out the sum of the interior angles of a pentagon.
The diagram shows a regular hexagon. Work out the size of one interior angle, marked x.
Work out the size of each exterior angle of a regular octagon.
The exterior angle of a regular polygon is 24°. How many sides does the polygon have?
Each interior angle of a regular polygon is 150°. How many sides does the polygon have?
The interior angles of a polygon add up to 1080°. How many sides does the polygon have?
Avoid these
The mistakes students make most often
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 students and parents ask
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