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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn4MA1Papers 1H, 2H · Higher tierGeometry and trigonometryAngles and polygons
Geometry and trigonometry · Subtopic 1 of 13

Angles and Polygons

A handful of angle facts (straight lines 180∘180^\circ, points 360∘360^\circ, triangles 180∘180^\circ, and the three parallel-line pairs) solve almost every angle problem. For polygons, start with the exterior angles: they always add up to 360∘360^\circ.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27
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Section 1 of 14

Learning objectives

  • Use angles on a straight line, at a point, and vertically opposite angles.
  • Use corresponding, alternate and co-interior angles with parallel lines.
  • Use the angle sums of triangles and quadrilaterals, and the exterior angle of a triangle.
  • Find interior and exterior angles of polygons, and the number of sides of a regular polygon.
Pause and think

Walk all the way around the edge of any polygon, turning at each corner. By the time you are back where you started, how far have you turned in total?

Section 2 of 14

Core idea

Start hereWhy do hexagons tile a floor?

Bathroom tiles are often squares or hexagons, never regular pentagons. The reason is angles: at each corner the tiles must fit exactly into 360∘360^\circ. A few simple angle facts explain which shapes work.

The big idea

Straight line 180∘180^\circ; point 360∘360^\circ; triangle 180∘180^\circ. Parallel lines: alternate and corresponding angles equal, co-interior add to 180∘180^\circ. Polygon: interior sum (n−2)×180∘(n-2)\times180^\circ; exterior angles sum to 360∘360^\circ.

1. Basic facts

  • Straight line 180∘180^\circ; round a point 360∘360^\circ.
  • Vertically opposite angles are equal.
  • Triangle 180∘180^\circ; exterior angle = sum of the two opposite interior angles.
  • Isosceles: base angles equal.
Triangle angles add to 180°
  1. draw a line through the top vertex parallel to the base\text{draw a line through the top vertex parallel to the base}
    Construction.
  2. left angle=A, right angle=B\text{left angle}=A,\ \text{right angle}=B
    Alternate angles with the base angles.
  3. A+C+B=180∘A+C+B=180^\circ
    The three angles at the top lie on a straight line.

2. Parallel lines

Parallel lines cut by a transversal
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The two lower-left/upper-right angles are alternate (Z). The angle at the top on the same side as the bottom one is corresponding (F). All three marked angles are equal.
  • Alternate (Z-shape): equal.
  • Corresponding (F-shape): equal.
  • Co-interior (C-shape): add to 180∘180^\circ.

3. Polygons

A hexagon split into 4 triangles: 4 × 180° = 720°
From one corner, a polygon with nn sides splits into n−2n-2 triangles.
Interior and exterior sums
  1. n-gon→(n−2) trianglesn\text{-gon}\to(n-2)\text{ triangles}
    Join one vertex to every non-adjacent vertex.
  2. interior sum=(n−2)×180∘\text{interior sum}=(n-2)\times180^\circ
    Each triangle contributes 180∘180^\circ.
  3. exterior sum=n×180∘−(n−2)180∘=360∘\text{exterior sum}=n\times180^\circ-(n-2)180^\circ=360^\circ
    Each interior + exterior pair is 180∘180^\circ.
Worked exampleRegular polygons

(a) Angles of a regular hexagon. (b) Sides if each exterior angle is 24∘24^\circ. (c) Sides if each interior angle is 156∘156^\circ.

Goal
Angles; nn.
Strategy
Use exterior angles: 360n\frac{360}n.
  1. ext=60∘, int=120∘, sum=720∘\text{ext}=60^\circ,\ \text{int}=120^\circ,\ \text{sum}=720^\circ
    360÷6360\div6; 180−60180-60; 4×1804\times180.
  2. n=360÷24=15n=360\div24=15
    Exterior angles sum to 360∘360^\circ.
  3. ext=180−156=24∘⇒n=15\text{ext}=180-156=24^\circ\Rightarrow n=15
    Convert to exterior first.
Answer120∘120^\circ each; 15 sides; 15 sides.
✓ Check(15−2)180/15=156∘(15-2)180/15=156^\circ ✓.
Exam tipGive reasons Write the reason each time: "alternate angles", "angles in a triangle", etc.
Pause and think

Can a regular polygon have interior angle 100∘100^\circ?

Summary

  • Line 180, point 360, triangle 180.
  • Z, F equal; C supplementary.
  • (n−2)180(n-2)180; exterior 360.
Key words
Alternate angles
Equal angles in a Z shape.
Exterior angle
Angle between one side and the extension of the next.
HistoryWhy 360? The Babylonians counted in base 6060, and their astronomers noticed the Sun moves about one part in 360360 of its yearly path each day. Dividing a circle into 360360 degrees, each split into 6060 minutes of 6060 seconds, comes from them, nearly 40004000 years ago. And 360360 divides neatly by 2,3,4,5,6,8,9,10,12,…2,3,4,5,6,8,9,10,12,\ldots
Fun factBees chose hexagons. Only three regular polygons tile a flat surface with no gaps: triangles, squares and hexagons, because their interior angles (60∘60^\circ, 90∘90^\circ, 120∘120^\circ) divide 360∘360^\circ exactly. Of the three, the hexagon encloses the most area for the least wall, which is why honeycomb uses the least wax.
Section 3 of 14

Key formulae

Sum of interior angles(n−2)×180∘(n-2)\times180^\circ
Sum of exterior angles360∘ (any polygon)360^\circ\ \text{(any polygon)}
Regular polygonexterior=360∘n,interior=180∘−exterior\text{exterior}=\frac{360^\circ}{n},\qquad\text{interior}=180^\circ-\text{exterior}
TrickExterior first, always. For regular polygons, the exterior angle is the easy one. Number of sides =360÷=360\div exterior angle. Given an interior angle of 156∘156^\circ? The exterior is 24∘24^\circ, so n=360÷24=15n=360\div24=15.
Section 4 of 14

Worked examples

Example 1 · Triangle with algebra0 / 2 steps
The angles of a triangle are xx, x+20∘x+20^\circ and 2x2x. Find xx.
    Example 2 · Regular polygon0 / 2 steps
    Each interior angle of a regular polygon is 150∘150^\circ. How many sides does it have?
      Section 5 of 14

      Explore the graph

      Interior angle of a regular polygon
      Loading the interactive graph…
      Try this
      1. The point shows the interior angle of a regular polygon with ss sides. A hexagon (s=6s=6) gives 120∘120^\circ.
      2. Increase ss. The angle gets closer to 180∘180^\circ but never reaches it. Why?
      3. Which values of ss give a whole number of degrees? (Hint: ss must divide 360360.)

      Interior angle =180−360s=180-\dfrac{360}s: as ss grows, the polygon looks more like a circle.

      In the real worldCoins with corners. The UK 5050p and 2020p coins are curved heptagons: regular seven-sided shapes with bulging sides that make them the same width in every direction, so vending machines can measure them however they roll.
      Section 6 of 14

      Common mistakes

      Incorrect"Z-angles" as a reason
      Correct"Alternate angles"
      Examiners want the proper names.
      IncorrectCo-interior angles are equal
      CorrectThey add to 180∘180^\circ
      Only alternate and corresponding angles are equal.
      IncorrectInterior angle =360n=\dfrac{360}n
      CorrectThat is the exterior angle
      Interior =180∘−360∘n=180^\circ-\dfrac{360^\circ}n.
      IncorrectAngle sum =n×180∘=n\times180^\circ
      Correct(n−2)×180∘(n-2)\times180^\circ
      An nn-gon splits into n−2n-2 triangles.
      Me

      the interior angle of a regular hexagon is 60∘60^\circ

      Exterior angle

      I'm 60∘60^\circ

      Interior angle

      I'm 120∘120^\circ. We add to 180∘180^\circ, remember?

      Section 7 of 14

      Exam tip

      Exam tipDiagrams are not to scale. Never measure an angle on an exam diagram, and never assume an angle is 90∘90^\circ or that lines are parallel unless you are told. Write each step with its reason, e.g. "x=65∘x=65^\circ (alternate angles)".
      Section 8 of 14

      Guided practice

      SA-4024-GE-G012 marksL1 FoundationNon-calculatorPolygons
      Find the size of one interior angle of a regular hexagon.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-4024-GE1-0012 marksL1 FoundationNon-calculatorAngle facts
      Three angles on a straight line are xx, 2x2x and 3x3x. Find xx.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-GE-0041 markL1 FoundationNon-calculatorAngle facts
      Two parallel lines are crossed by a transversal. One of a pair of co-interior angles is 115∘115^\circ. Find the other.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-GE-0032 marksL1 FoundationNon-calculatorAngle facts
      Triangle ABCABC is isosceles with AB=ACAB=AC, and angle BAC=52∘BAC=52^\circ. Find angle ABCABC.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-GE1-0052 marksL2 StandardNon-calculatorAngle facts
      The angles of a triangle are xx, x+20∘x+20^\circ and 2x2x. Find the largest angle.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-GE1-0032 marksL1 FoundationNon-calculatorPolygons
      Find the size of each exterior angle of a regular octagon.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-GE-0012 marksL2 StandardNon-calculatorPolygons
      Each exterior angle of a regular polygon is 24∘24^\circ. How many sides does it have?

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-GE-0022 marksL1 FoundationNon-calculatorPolygons
      Find the sum of the interior angles of a nonagon (99 sides).

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-GE1-0043 marksL2 StandardNon-calculatorPolygons
      Four angles of a pentagon are 100∘100^\circ, 110∘110^\circ, 120∘120^\circ and 90∘90^\circ. Find the fifth angle.

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      Salam Academy Original Practice · Proposed mark scheme

      More questions on every skill in this subtopic are in the question bank under Practise.

      Section 10 of 14

      Challenge

      SA-4024-GE-0164 marksL4 ChallengingNon-calculatorPolygons
      Each interior angle of a regular polygon is 44 times each exterior angle. (i) How many sides does the polygon have? (ii) Find the sum of its interior angles.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • Straight line 180∘180^\circ, point 360∘360^\circ, vertically opposite angles equal.
      • Parallel lines: corresponding and alternate equal, co-interior add to 180∘180^\circ.
      • Triangle 180∘180^\circ; exterior angle == sum of the two opposite interior angles.
      • Polygon: interior sum (n−2)×180∘(n-2)\times180^\circ, exterior sum 360∘360^\circ.
      Section 12 of 14

      Quick quiz

      Q1 · SA-4024-GE1-0021 markL1 FoundationCalculatorAngle facts
      A line crosses two parallel lines, making a Z-shape. The two angles inside the Z are
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Q2 · SA-4024-GE1-0061 markL1 FoundationCalculatorPolygons
      The interior angles of any quadrilateral add up to
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Q3 · SA-4024-GE-0171 markL1 FoundationNon-calculatorAngle facts
      In triangle ABCABC, angle A=58∘A=58^\circ and angle B=47∘B=47^\circ. Side BCBC is extended beyond CC to DD. Find angle ACDACD.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      International GCSE Mathematics A 4MA1
      Where
      Geometry and trigonometry
      Expected
      The same angle facts and polygon angles, with reasons on Higher tier.
      Calculator
      Calculator allowed in both papers.
      Typical style
      Similar question styles; some content is Higher tier only.
      Open in
      4MA1 · Papers 1H, 2H4MA1 · Papers 1F, 2F
      Section 14 of 14

      Past paper questions

      7 questions (23 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

      1. 4024/12 · Oct/Nov 2025Question 34 marks

        An angle in a parallelogram using isosceles triangles.

        Question paper, page 4Mark scheme
      2. 4024/22 · Oct/Nov 2025Question 93 marks

        An angle where a pentagon meets a hexagon.

        Question paper, page 6Mark scheme
      3. 4024/12 · May/June 2025Question 43 marks

        Angle reasoning with isosceles triangles and straight lines.

        Question paper, page 5Mark scheme
      4. 4024/22 · May/June 2025Question 42 marks

        Find two unknown angles from a diagram.

        Question paper, page 4Mark scheme
      5. 4024/22 · May/June 2025Question 82 marks

        Interior angle of a regular octagon.

        Question paper, page 6Mark scheme
      6. 4024/12 · Oct/Nov 2024Question 152 marks

        Interior angles of regular polygons.

        Question paper, page 8Mark scheme
      Show 1 more
      1. 4024/22 · May/June 2024Question 107 marks

        Interior angles of polygons and trigonometry.

        Question paper, page 16Mark scheme

      All 4024 past paper questions by topic

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 4MA1, 2016 specification (official specification).