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Polygons

Polygons

Angles in polygons

Formulas Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Interior and exterior angles
  4. Ways they test it
  5. Mistakes to avoid
  6. Summary
  7. Practice
  8. Can I do this?
  9. Quiz

In this note

Syllabus 4.6: angles of polygons
  • 4.6Use the interior angle sum \((n-2)\times180^\circ\) and the exterior angle sum 360°.
  • 4.6Find the interior and exterior angles of a regular polygon, and the number of sides.
  • 4.6Solve problems with irregular polygons and polygons meeting at a point.

Before you start: Angle facts.

Formula bank

Interior angle sumLearn it\((n-2)\times180^\circ\)n = number of sides; any polygon.Use it when the total of the inside angles.
Exterior angle sumLearn it\(360^\circ\)any polygon.Use it when exterior angles.
Regular polygonLearn itexterior \(=\dfrac{360^\circ}n\); interior \(=180^\circ-\) exteriorbackwards: \(n=\dfrac{360^\circ}{\text{exterior}}\).Use it when all sides and angles equal.

Interior and exterior angles

A pentagon splits into 3 triangles123
Diagonals from one corner split a pentagon into 3 triangles.

An n-sided polygon splits into \(n-2\) triangles, so its angles add to \((n-2)\times180^\circ\): pentagon 540°, hexagon 720°, octagon 1080°.

Interior and exterior angle of a regular hexagon120°60°
Regular hexagon: exterior 60°, interior 120°.

Walking round any polygon you turn through one full turn, so the exterior angles add to 360°. At each corner interior + exterior = 180°.

General methodHow to answer any polygon-angle question

  1. Regular polygon, n known: exterior = 360 ÷ n, then interior = 180 − exterior.
  2. Regular polygon, angle known: exterior = 180 − interior, then n = 360 ÷ exterior (must be a whole number).
  3. Irregular polygon with algebra: set the sum of the angles = \((n-2)\times180\) and solve.
  4. Shapes meeting at a point: their angles add to 360°.

Example Each interior angle of a regular polygon is 168°. How many sides?

Exterior first, then 360 ÷ exterior.

  1. exterior = 180 − 168 = 12°
  2. n = 360 ÷ 12

Answer 30 sides

Try it yourself

1
Interior angle of a regular decagon?
Show answer

Answer 144

2
Exterior angle 24°: number of sides?
Show answer

Answer 15

3
Interior angle sum of a heptagon?
Show answer

Answer 900

Ways they test it

Type 1 Number of sides

"The interior angle of a regular polygon is 165°. Find the number of sides." (2 marks)

Spot it an interior angle given.

  1. Exterior = 15°.
  2. 360 ÷ 15.

For example exterior 15°

    Answer 24 sides

    Type 2 Interior angle

    "Find the interior angle of a regular 24-sided polygon." (2 marks)

    Spot it a number of sides.

    1. 360 ÷ 24 = 15.
    2. 180 − 15.

    For example exterior 15°

      Answer 165°

      Type 3 Irregular with algebra

      "The angles of a pentagon are \((x+40)^\circ\), \(2x^\circ\), \(2x^\circ\), \((3x-10)^\circ\), 110°. Find x." (3 marks)

      Spot it a list of angles in x.

      1. Sum = (5 − 2) × 180 = 540.
      2. Solve.

      For example \(8x+140=540\)

        Answer x = 50

        Type 4 Polygons joined

        "A square and a regular hexagon share a side. Find the angle between them." (3 marks)

        Spot it two shapes meeting.

        1. Each interior angle.
        2. Angles at a point = 360°.

        For example 90° and 120°

          Answer 150°

          Mistakes to avoid

          • ✗ n = 360 ÷ interior✓ n = 360 ÷ exterior360° is the exterior sum.
          • ✗ Interior sum = n × 180✓ (n − 2) × 180There are n − 2 triangles.
          • ✗ Exterior sum depends on n✓ always 360°One full turn.

          Summary

          • Interior sum \((n-2)\times180^\circ\); exterior sum 360°.
          • Interior + exterior = 180° at each corner.
          • Regular: exterior = 360 ÷ n; n = 360 ÷ exterior.

          If you see …

          • "regular polygon" with an anglego through the exterior angle.
          • angles of a polygon in xsum = (n − 2) × 180.

          Practice

          Practice: easy, medium, hard

          Easy

          1
          Interior angle sum of an octagon?
          Show answer

          Answer 1080

          2
          Exterior angle of a regular hexagon?
          Show answer

          Answer 60

          3
          Interior angle of a regular pentagon?
          Show answer

          Answer 108

          Medium

          1
          Interior angle of a regular 24-sided polygon?
          Guide me

          What is it asking? an angle

          What do we know? exterior 360 ÷ 24

          Plan 180 − 15

          Show working
          1. 165°

          Answer 165

          2
          Each interior angle of a regular polygon is 168°. Number of sides?
          Guide me

          What is it asking? n

          What do we know? exterior 12°

          Plan 360 ÷ 12

          Show working
          1. 30

          Answer 30

          3
          A regular polygon has exterior angle 20°. Find its interior angle.
          Guide me

          What is it asking? an angle

          What do we know? 180 − exterior

          Plan 180 − 20

          Show working
          1. 160°

          Answer 160

          Hard

          1
          The angles of a pentagon are \((x+40)^\circ\), \(2x^\circ\), \(2x^\circ\), \((3x-10)^\circ\) and 110°. Find the largest angle.
          Guide me

          What is it asking? an angle

          What do we know? sum 540

          Plan \(8x+140=540\Rightarrow x=50\)

          Show working
          1. angles 90, 100, 100, 140, 110

          Answer 140

          PartAnswer / workingMarksGuidance
          \(8x+140=540\)M1
          x = 50A1
          140°A1
          Total: 3 marks
          2
          A square and a regular hexagon share one side. Find the angle between them at a shared corner, outside both.
          Guide me

          What is it asking? an angle

          What do we know? hexagon 120°, square 90°

          Plan 360 − 210

          Show working
          1. 150°

          Answer 150

          PartAnswer / workingMarksGuidance
          120B1
          360 − 90 − 120M1
          150A1
          Total: 3 marks
          3
          Can a regular polygon have interior angles of 100°? Type yes or no.
          Guide me

          What is it asking? yes or no

          What do we know? exterior would be 80°

          Plan 360 ÷ 80 = 4.5

          Show working
          1. 4.5 sides is impossible

          Answer no

          PartAnswer / workingMarksGuidance
          no, with reasonB1
          Total: 1 mark

          Can I do this?

          Anything unticked: revise that section first. All ticked: take the quiz.

          Quiz

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