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Maths · Angles and trigonometry

Exterior angles of a polygon

Walk all the way round any polygon and you turn through exactly one full turn. So the exterior angles always add up to \(360^\circ\) - the quickest route to the angles of a regular polygon.

  • 4 key terms
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Last Lesson and Before

Answer each one, then check.

  1. 1

    Last lesson: what do the interior angles of a pentagon add up to?

    Show answerHide answer

    \(540^\circ\)

  2. 2

    What do angles on a straight line add up to?

    Show answerHide answer

    \(180^\circ\)

  3. 3

    Work out \(360 \div 8\).

    Show answerHide answer

    45

  4. 4

    What is each interior angle of a regular hexagon?

    Show answerHide answer

    \(120^\circ\)

Learning Objectives

  1. 1Know that the exterior angles of any polygon add up to \(360^\circ\).
  2. 2Find the exterior angle of a regular polygon.
  3. 3Use interior angle + exterior angle = \(180^\circ\).
  4. 4Find the number of sides of a regular polygon from one of its angles.

Exterior Angle Facts

Two facts do almost all the work.

  • The sum

    The exterior angles of any polygon add up to \(360^\circ\).

  • Interior and exterior

    At each vertex, interior angle + exterior angle = \(180^\circ\) (angles on a straight line).

  • Regular polygon

    Each exterior angle \(= \dfrac{360}{n}\).

  • Number of sides

    \(n = \dfrac{360}{\text{exterior angle}}\).

Angles of a Regular Pentagon

Work out the exterior and interior angles of a regular pentagon.

Show the solutionHide the solution
  1. 1 Exterior angle \(360 \div 5 = 72^\circ\)
  2. 2 Interior angle: they add up to \(180^\circ\) \(180 - 72 = 108^\circ\)
  3. 3 Check with last lesson's formula \(540 \div 5 = 108^\circ\)

AnswerExterior \(72^\circ\), interior \(108^\circ\)

How Many Sides?

Each interior angle of a regular polygon is \(160^\circ\). How many sides does it have?

Show the solutionHide the solution
  1. 1 Find the exterior angle \(180 - 160 = 20^\circ\)
  2. 2 Exterior angles add up to \(360^\circ\) \(360 \div 20 = 18\)

Answer18 sides

Which Method?

Exterior angles are usually quicker for regular polygons.

  1. 1 Regular, angles wanted

    Exterior \(= 360 \div n\), then interior \(= 180 -\) exterior.
  2. 2 Regular, sides wanted

    Exterior \(= 180 -\) interior, then \(n = 360 \div\) exterior.
  3. 3 Irregular

    Use the angle sum \((n - 2) \times 180\), or exterior angles adding up to \(360^\circ\).
  4. 4 Shapes joined together

    Use angles around a point: they add up to \(360^\circ\).

Polygons Meeting at a Point

A regular hexagon and a square share a side, and meet at a point. Work out the angle \(x\) in the gap between them at that point.

Show the solutionHide the solution
  1. 1 Interior angle of a regular hexagon \(180 - 360 \div 6 = 120^\circ\)
  2. 2 Interior angle of a square \(90^\circ\)
  3. 3 Angles around a point add up to \(360^\circ\) \(x = 360 - 120 - 90 = 150\)

Answer\(x = 150^\circ\)

Impossible Polygons

For each angle, decide whether it could be the exterior angle of a regular polygon. If it could, say how many sides the polygon has: \(40^\circ\), \(50^\circ\), \(24^\circ\), \(70^\circ\), \(15^\circ\), \(1^\circ\).

1. Divide 360 by the angle.

2. A whole number means it is possible.

3. Write down the number of sides.

A good answer shows: \(40^\circ\): 9 sides. \(50^\circ\): no (7.2). \(24^\circ\): 15 sides. \(70^\circ\): no. \(15^\circ\): 24 sides. \(1^\circ\): 360 sides. An exterior angle works only if it divides exactly into 360.

Can I...?

  1. 1Recall that exterior angles add up to \(360^\circ\).
  2. 2Find the exterior angle of a regular polygon.
  3. 3Use interior + exterior = \(180^\circ\).
  4. 4Find the number of sides from an angle.
  5. 5Decide whether an angle is possible for a regular polygon.
  6. 6Solve problems with polygons meeting at a point.

Summary & Exam Focus

  • Exterior angles of any polygon add up to \(360^\circ\).
  • Regular polygon: exterior angle \(= 360 \div n\), and \(n = 360 \div\) exterior angle.
  • Interior + exterior \(= 180^\circ\).

Exam focus

The diagram shows a regular hexagon and a square that share a side. Work out the size of angle \(x\). (3 marks) (3 marks)

For any regular polygon question, work out the exterior angle first: \(360 \div n\). Everything else follows from it.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Exterior angle
The angle between one side of a polygon and the extension of the side next to it.
Interior angle
The angle inside a polygon at a vertex.
Angles around a point
Angles meeting at a point; they add up to \(360^\circ\).
Regular polygon
A polygon with all sides and all angles equal.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 2 marks

    Work out the size of each exterior angle of a regular octagon.

    Show answerHide answer

    Model answer

    \(360 \div 8 = 45^\circ\)

    Mark scheme

    • \(360 \div 8\) — M1
    • \(45^\circ\) — A1
  2. Question 2 Non-calculator 2 marks

    Each interior angle of a regular polygon is \(160^\circ\). Work out the number of sides of the polygon.

    Show answerHide answer

    Model answer

    Exterior angle \(180 - 160 = 20^\circ\). \(360 \div 20 = 18\) sides.

    Mark scheme

    • \(180 - 160 = 20\) — M1
    • 18 — A1
  3. Question 3 Non-calculator 3 marks

    The diagram shows a regular hexagon and a square. They share a side. Work out the size of angle \(x\).

    A regular hexagon and a square sharing a vertical side, with the angle x marked outside both shapes at the top of the shared side.
    Show answerHide answer

    Model answer

    Interior angle of the hexagon: \(180 - 360 \div 6 = 120^\circ\). Interior angle of the square: \(90^\circ\). Angles around a point: \(x = 360 - 120 - 90 = 150^\circ\).

    Mark scheme

    • \(120^\circ\) for the hexagon's interior angle — M1
    • \(360 - 120 - 90\) — M1
    • \(150^\circ\) — A1
  4. Question 4 Non-calculator 2 marks

    Ali says, "I have drawn a regular polygon with exterior angles of \(50^\circ\)." Explain why Ali cannot be right.

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    Model answer

    \(360 \div 50 = 7.2\). The number of sides must be a whole number, so no regular polygon has exterior angles of \(50^\circ\).

    Mark scheme

    • \(360 \div 50\) — M1
    • 7.2 and a statement that it is not a whole number — C1

Quick check

  1. What do the exterior angles of a decagon add up to?

    1. A\(1440^\circ\)
    2. B\(360^\circ\)
    3. C\(180^\circ\)
    4. D\(36^\circ\)
    Show answerHide answer

    B: \(360^\circ\)

    The exterior angles of every polygon add up to \(360^\circ\).

  2. A regular polygon has exterior angles of \(30^\circ\). How many sides does it have?

    1. A6
    2. B10
    3. C30
    4. D12
    Show answerHide answer

    D: 12

    \(360 \div 30 = 12\).

  3. Each interior angle of a regular polygon is 4 times its exterior angle. How many sides does it have?

    1. A5
    2. B8
    3. C10
    4. D36
    Show answerHide answer

    C: 10

    Exterior \(e\), interior \(4e\): \(e + 4e = 180\), so \(e = 36\) and \(n = 360 \div 36 = 10\).

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