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Angles in polygons Cambridge IGCSE Mathematics revision

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In plain words

A polygon is any shape with straight sides. However many sides it has, its outside angles always add up to a full turn, 360°. That one fact answers most polygon questions.

Three things to know

  1. The exterior angles of any polygon add up to 360°. In a regular polygon they are all equal, so each one is 360° ÷ the number of sides.
  2. An interior angle and its exterior angle lie on a straight line, so they add up to 180°.
  3. The interior angles of a polygon with n sides add up to (n − 2) × 180°.

Worked example

Find the size of each interior angle of a regular octagon.

  1. An octagon has 8 sides, so each exterior angle is 360 ÷ 8 = 45°.
  2. Interior + exterior = 180°.
  3. Each interior angle is 180 − 45 = 135°.

Tips and tricks

  • Go through the exterior angle: it's nearly always quicker than the (n − 2) × 180 formula.
  • Number of sides = 360 ÷ exterior angle.
5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Angles in polygons: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

  1. What do the exterior angles of any polygon add up to?
    • 180°
    • 360° (the answer)
    • 540°
    • it depends on the number of sides

    Walking round the polygon you make one full turn.

  2. What is each exterior angle of a regular pentagon?
    • 60°
    • 72° (the answer)
    • 108°
    • 90°

    360 ÷ 5.

  3. What is each interior angle of a regular hexagon?
    • 60°
    • 108°
    • 120° (the answer)
    • 135°

    Exterior 60°, so interior 180 − 60.

  4. What do the interior angles of a pentagon add up to?
    • 360°
    • 540° (the answer)
    • 720°
    • 900°

    (5 − 2) × 180.

  5. A regular polygon has exterior angles of 40°. How many sides does it have?
    • 8
    • 9 (the answer)
    • 10
    • 40

    360 ÷ 40.

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