Angles in Polygons
Module 4 · Lesson 4.03 · Spec G3, G4
Interior and exterior angles of regular and irregular polygons — why (n − 2) × 180° and 360° are all you really need.
Building on 4.01 & 4.02
Polygons are just triangles put together. Every polygon result in this lesson is derived from the triangle-sum fact (4.01) plus angles on a straight line (4.01). Parallel-line rules (4.02) show up in harder polygon problems when the figure contains parallel sides.
What's new here
Two key facts: interior angles of an n-sided polygon sum to (n − 2) × 180°, and exterior angles of any polygon sum to 360°.
What carries over
State a named reason after every angle you find. Write the value in a box; write the reason beside it.
Big Idea
Most polygon questions can be solved in two or three lines if you pick the right tool. The exterior-angle approach is usually faster than the interior formula when a regular polygon is involved.
Polygons: Regular vs Irregular
A polygon is a closed 2D shape made of straight sides. The number of sides is n; an n-sided polygon has n vertices and n interior angles.
Vocabulary
Regular polygon: all sides equal AND all angles equal. Examples: equilateral triangle, square, regular hexagon.
Irregular polygon: sides or angles differ. Most real-world shapes are irregular.
Standard names: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
Interior Angle Sum: (n − 2) × 180°
Pick any vertex of a polygon and draw every diagonal from it. An n-sided polygon splits into (n − 2) triangles. Each triangle's angles sum to 180°, so:
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Interior sum = (n − 2) × 180°
Why This Works (Derivation)
From one vertex, you can draw a diagonal to every other vertex except the two it's already joined to. That's n − 3 diagonals, which split the polygon into n − 2 triangles. Each triangle sums to 180°, so the interior sum is (n − 2) × 180°.
Checks
Triangle (n=3): (3−2) × 180 = 180° ✓
Quadrilateral (n=4): (4−2) × 180 = 360° ✓
Pentagon (n=5): (5−2) × 180 = 540°
Each extra side adds one triangle — 180° more.
Each Interior of a Regular Polygon
If the polygon is regular, every interior angle is equal. Divide the sum by n:
Each interior of a regular n-gon = (n − 2) × 180° / n
Quick Values (Regular)
Triangle 60°, square 90°, pentagon 108°, hexagon 120°, octagon 135°, decagon 144°. Worth memorising the hexagon (120°) and octagon (135°) — they appear on every paper.
Exterior Angles Sum to 360°
At each vertex of a polygon, the exterior angle is the angle you turn through when walking along the outside. For any polygon (regular or irregular, convex), going all the way around is exactly one full turn.
Sum of exterior angles = 360° (always)
Key Fact
The exterior sum is always 360° — regardless of how many sides or whether the polygon is regular. For a regular n-gon, each exterior = 360°/n.
Interior + Exterior = 180°
At every vertex, the interior and exterior angles sit on a straight line — the side of the polygon extended. So they always sum to 180°.
At each vertex: interior + exterior = 180°
Why This Matters
This bridge lets you switch instantly between interior and exterior angles. If you know one, subtract from 180° to get the other. AQA problems often give the interior angle and the quickest route to the answer uses the exterior.
Worked Example — Interior Sum01
Find the sum of the interior angles of a heptagon (7-sided polygon). [1 mark]
M
Method: Use the interior sum formula (n − 2) × 180° with n = 7.
C
Calculate: (7 − 2) × 180 = 5 × 180 = 900°
E
Evaluate / Answer: The interior angles of a heptagon sum to 900°.
Mark split: B1 for 900°. No reason mark is typically required for a pure "interior sum" question.
Worked Example — Each Interior Angle02
Work out the size of each interior angle of a regular octagon. Give a reason. [2 marks]
M
Method (fast route): Use exterior = 360°/n, then interior = 180° − exterior. Faster than the interior formula.
C
Calculate: exterior = 360 / 8 = 45°. Interior = 180 − 45 = 135°.
E
Evaluate / Answer: each interior angle = 135° (exterior angles of a regular polygon sum to 360°; interior and exterior angles on a straight line sum to 180°).
Alt. method: (8 − 2) × 180 / 8 = 1080/8 = 135°. Same answer, slightly more arithmetic.
Worked Example — Number of Sides03
Each interior angle of a regular polygon is 156°. Work out the number of sides. [3 marks]
M
Method: Work from the exterior angle. Find exterior = 180° − interior. Then n = 360° / exterior.
C
Calculate: exterior = 180 − 156 = 24°. n = 360 / 24 = 15.
E
Evaluate / Answer: the polygon has 15 sides (a pentadecagon). Check: (15 − 2) × 180 / 15 = 2340 / 15 = 156° ✓.
Mark split: M1 method (exterior = 180 − 156 OR interior formula), A1 exterior = 24, A1 n = 15. The exterior-angle approach is the one AQA expects for "find n" questions.
Worked Example — Irregular Polygon04
ABCDE is a pentagon. Angle A = 130°, angle B = 118°, angle C = 95°, angle D = 102°. Find angle E. Give a reason. [3 marks]
M
Method: Interior sum of a pentagon is (5 − 2) × 180 = 540°. Angle E = 540° minus the four given angles.
C
Calculate: sum so far = 130 + 118 + 95 + 102 = 445°. Angle E = 540 − 445 = 95°.
E
Evaluate / Answer: angle E = 95° (interior angles of a pentagon sum to 540°).
Mark split: M1 for using the 540° sum, M1 for subtracting the four given angles, A1 for 95°. Writing "540" earns the first mark even if later arithmetic slips.
Common Polygons at a Glance
Memorising the values for the polygons below saves time in every exam.
Sides (n)NameInterior sumEach interior (regular)Each exterior (regular)
3Triangle180°60°120°
4Quadrilateral360°90°90°
5Pentagon540°108°72°
6Hexagon720°120°60°
7Heptagon900°≈128.57°≈51.43°
8Octagon1080°135°45°
9Nonagon1260°140°40°
10Decagon1440°144°36°
12Dodecagon1800°150°30°
Pattern
Each extra side adds 180° to the interior sum. The exterior angle always divides 360 exactly — so the "nice" regular polygons have n that divides 360 (3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24…). The heptagon (n=7) is awkward because 360 ÷ 7 isn't a whole number.
Accepted Reason Wording
Polygon questions often carry a reason mark (B1 or M1). AQA mark schemes accept these phrasings:
When you useWrite this (scores reason mark)
Interior sum (specific polygon)"angles in a pentagon sum to 540°" — name the polygon, state the value
Interior sum (general)"interior angles of an n-sided polygon sum to (n − 2) × 180°"
Exterior sum of any polygon"exterior angles of a polygon sum to 360°"
Interior + exterior at a vertex"angles on a straight line sum to 180°"
Regular polygon — each equal"regular polygon" (AQA accepts this on its own when dividing by n)
AQA Mark-Scheme Rule
You need to state the rule AND the numerical value. Writing "angles in a pentagon sum to something" without 540° loses the reason mark. Always write the number: 180°, 360°, 540°, 720°, 1080° etc.
"Z angles" equivalents
Just as AQA refuses "Z angles" for alternate (lesson 4.02), informal polygon wording doesn't score. "Polygon angles add up" is not enough — name the polygon OR state (n − 2) × 180°.
Examiner Alerts
Alert 1 — Wrong formula for exterior sum
Students sometimes write "exterior sum = (n − 2) × 180°". Wrong. That's the interior sum. Exterior sum is always 360°, regardless of n.
Alert 2 — Mixing interior and exterior at a vertex
"Interior angle of a regular hexagon = 360° / 6 = 60°". That's the exterior angle. The interior is 180 − 60 = 120°. Double-check which one the question wants.
Alert 3 — Using interior formula when exterior is faster
For "find n given an interior angle", using (n − 2) × 180 / n = given angle leads to messy algebra. The exterior route — exterior = 180 − interior, then n = 360 / exterior — is almost always quicker and usually what AQA expects.
Alert 4 — Assuming a polygon is regular
If the question says "pentagon" but not "regular pentagon", the interior angles may differ. You can use the sum (540°), but you cannot divide by 5 to get each angle. Check the wording before dividing.
Strategy: Which Formula?
Pick the right tool for the question. Three scenarios cover nearly everything:
A. Given n — find interior sum
Use (n − 2) × 180°. One-step. Example: hexagon → (6 − 2) × 180 = 720°.
B. Given n, regular — find each interior or each exterior
Fastest: each exterior = 360°/n, then interior = 180 − exterior.
Example: regular decagon → exterior = 36°, interior = 144°.
C. Given an interior angle of a regular polygon — find n
Exterior = 180° − interior. n = 360° / exterior.
Example: interior = 144° → exterior = 36° → n = 10.
When to Use Interior Formula
Reach for (n − 2) × 180 when you need the total sum of angles in an irregular polygon (WE4 style). Avoid it for "find n given a regular interior" — exterior is cleaner.
Multi-Step: Combining Polygon Facts
Harder questions combine the polygon rules with 4.01 / 4.02 facts. A regular polygon joined to an equilateral triangle or square is the classic AQA pattern.
Typical Chained Question
A regular hexagon and an equilateral triangle share a side. At the shared vertex, show that the outer hexagon edge and the outer triangle edge lie on a straight line.
M. Find the interior angle of each polygon at the shared vertex. If they sum to 180°, the two outer edges form a straight line (because together the two interior angles fill 180° on one side of the shared edge).
C. Regular hexagon: exterior = 360 ÷ 6 = 60°, so interior = 180 − 60 = 120°.
Equilateral triangle: each angle = 60°.
Sum at shared vertex (hexagon interior + triangle interior) = 120° + 60° = 180°.
E.The two outer edges lie on a straight line. The two interior angles (120° and 60°) together fill 180°, so the remaining two edges must be collinear (angles on a straight line sum to 180°; regular hexagon each interior = 120°; equilateral triangle each angle = 60°).
General Chain
1
Identify n and whether each polygon is regular.
2
Find each interior angle using the polygon rule.
3
At a shared vertex or point, use angles at a point (360°) or angles on a straight line (180°) .
4
If the figure contains a triangle, bring in the triangle-sum (180°). If parallel lines, bring in alternate / corresponding / co-interior (4.02).
5
Write a named reason for every angle you find.
Write on the Diagram
Mark every angle you find directly on the figure with its reason beside. AQA mark schemes accept on-diagram working for method marks — often a lifeline if a later step goes wrong.
3-Letter Notation (from 4.01)
Use "angle ABC = …" rather than "x = …" when the diagram labels vertices. It tells the examiner exactly which angle you mean.
Summary
The two polygon facts
Interior angles sum = (n − 2) × 180° for any n-sided polygon.
Exterior angles sum = 360° for any polygon (regular or irregular).
Bridge between them
At each vertex: interior + exterior = 180° (angles on a straight line).
Regular polygon: each interior = (n − 2) × 180° / n; each exterior = 360° / n.
Why this matters
Polygon questions appear on nearly every GCSE paper, often combined with triangle-sum or parallel-line rules.
The exterior-angle approach saves time on "find n" problems.
The next lesson (4.04) switches to constructions, loci and bearings — ruler-and-compass work.
Never Forget
Exterior sum = 360°, always. Interior sum = (n − 2) × 180°. These two facts plus angles on a straight line (180°) solve the vast majority of polygon questions.
Checkpoint
You should now be able to
State and use the interior angle sum formula (n − 2) × 180° for any polygon.
State and use the fact that exterior angles of any polygon sum to 360°.
Find each interior or each exterior angle of a regular polygon.
Find the number of sides of a regular polygon given an interior or exterior angle.
Find a missing angle in an irregular polygon using the interior sum.
Choose the faster route (exterior) over messy interior algebra when finding n.
Test yourself in under two minutes
1
Interior angle sum of a nonagon (9-sided).
2
Each interior angle of a regular dodecagon (12-sided).
3
The interior angle of a regular polygon is 150°. How many sides?
4
Four angles of a pentagon are 100°, 120°, 95°, 110°. Find the fifth.
Answers: 1) (9−2)×180 = 1260°. 2) exterior = 360/12 = 30°; interior = 150°. 3) exterior = 30°; n = 360/30 = 12 sides. 4) 540 − (100+120+95+110) = 540 − 425 = 115°.
Next Lesson (4.04)
We switch to constructions, loci, bearings & scale drawings — ruler-and-compass work, where your arcs are the evidence of your method and must NOT be erased.
Next: Building on 4.01 & 4.02