Interior and exterior polygon angles
Use polygon angle sums and regularity, keeping equal-angle assumptions explicit.
Core and Extended: angle properties of regular polygons. Extended additionally applies these relationships to irregular polygons; that practice is labelled below.
Before you begin
- Use triangle angle sums.
- Solve a simple equation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How many triangles can a convex pentagon be split into from one vertex?
What you will learn
- Use (n − 2) × 180° for a simple polygon's interior sum.
- Find regular polygon angles and number of sides.
- For Extended, calculate missing angles in irregular polygons.
Build the interior sum from triangles
A convex n-sided polygon can be split from one vertex into n − 2 triangles, each with angle sum 180°. The polygon's interior sum is therefore (n − 2) × 180°. For a hexagon the sum is 4 × 180 = 720°. This sum applies to simple polygons; a self-crossing star is not treated by this elementary rule.
For a regular polygon, equal angles allow division by n. A regular hexagon has each interior angle 720/6 = 120°. Without regularity or another equal-angle condition, dividing the sum by n gives only an average, not every angle. Extended: for an irregular pentagon with four angles 100°, 110°, 120° and 95°, the fifth is 540 − 425 = 115°.
Exterior turns return to the starting direction
For a convex polygon, choose the exterior turning angle at each vertex in the same direction. The total is 360°, whatever the number of sides. A regular polygon has equal exterior angles, each 360°/n. Each interior angle and its adjacent exterior angle sum to 180°.
A regular polygon with exterior angle 24° has n = 360/24 = 15 sides and interior angle 156°. If an interior angle is given first, subtract it from 180° to obtain the exterior angle before finding n. The number of sides must be a whole number at least three; a result such as 7.5 signals an incompatible regular polygon or an arithmetic mistake.
Pause and explain
A regular octagon has what exterior angle?
Worked example
A regular polygon has interior angle 156°. Find its exterior angle and number of sides.
Show the worked solution
- The adjacent exterior angle is 180 − 156 = 24°.
- Equal exterior turns total 360°, so n = 360/24 = 15.
- Check: (15 − 2) × 180/15 = 156° per interior angle.
Answer Exterior 24°; 15 sides.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the interior sum and each interior angle of a regular decagon.
Give me a hint
Use n = 10 and divide only because the polygon is regular.
Compare my reasoning
- The sum is (10 − 2) × 180 = 1440°.
- Each interior angle is 1440/10 = 144°.
- Its exterior angle is 36°, and 144 + 36 = 180° checks the result.
Sum 1440°; each interior angle 144°.
Look for these in your work
- Used n − 2 for the sum.
- Justified division by equal angles.
A regular polygon has exterior angle 30°. Find its number of sides and each interior angle.
Give me a hint
Use the full exterior turn, not the interior sum, to find n.
Compare my reasoning
- n = 360/30 = 12 sides.
- Each interior angle is 180 − 30 = 150°.
- Check (12 − 2) × 180/12 = 150°.
12 sides; interior angle 150°.
Look for these in your work
- Used the correct exterior total.
- Checked the supplementary angle.
Extended: an irregular hexagon has five interior angles 110°, 125°, 130°, 105° and 120°. Find its sixth angle.
Give me a hint
The total still applies, but the angles are not assumed equal.
Compare my reasoning
- A hexagon's sum is (6 − 2) × 180 = 720°.
- The five known angles total 590°.
- The sixth is 720 − 590 = 130°; no division by six is justified.
130° (Extended irregular polygon).
Look for these in your work
- Used the sum without assuming regularity.
- Included all five supplied angles.
Common mistakes
- Using n triangles instead of n − 2.
- Confusing an interior angle with an exterior turn.
- Assuming an irregular polygon's angles are all equal.
What can you explain now?
A regular polygon has exterior angle 40°. How many sides does it have?
Compare with the explanation
9
Its equal exterior turns total 360°, so the number of sides is 360/40 = 9.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, start with a regular polygon's interior angle and recover its number of sides. For Extended, also find one missing irregular angle.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources