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Angles in Polygons (AQA GCSE Maths)

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Angles in Polygons

What is a polygon?

  • A polygon is a flat (plane) shape with n straight sides
    • For example:
      • A triangle is a polygon with 3 sides
      • A quadrilateral polygon with 4 sides
      • A pentagon is a polygon with 5 sides
  • In a regular polygon all the sides are the same length and all the angles are the same
    • A regular polygon with 3 sides is an equilateral triangle
    • A regular polygon with 4 sides is a square

What are the sums of angles in polygons?

  • To be able to work with angles in polygons, you need to be able to find the sums of angles in polygons
  • To find the sum of the interior angles in a polygon of n sides, use the rule
    • SUM OF INTERIOR ANGLES = 180° × (n – 2)
      • because the polygon can be split into n -2 triangles
    • For regular polygons, this can then be divided by n to find each individual angle
  • The sum of the exterior angles in any polygon always add up to 360°
    • TOTAL OF EXTERIOR ANGLES = 360°
      • The exterior angles are the angles extended out from each side on a straight line
      • It is important to note that the interior and exterior angles for each side lie on a straight line so they add up to 180° 

Angles in a Hexagon, downloadable IGCSE & GCSE Maths revision notes

Interior & Exterior Angle for Regular Polygon, IGCSE & GCSE Maths revision notes

Examiner Tip

  • Make sure you identify whether you are dealing with a regular or irregular polygon before you start a question
    • If the polygon is regular you may not be given any of the angles, but you can still find their sizes
    • If the polygon is irregular there will be more information given about the individual angles 

Worked example

The diagram below shows an irregular pentagon. 

Work out the value of x.

Polygon-Example-REWRITTEN-5-3-20, IGCSE & GCSE Maths revision notes

An irregular pentagon has 5 sides and 5 angles that are not all equal (2 or more may still be equal).

Use the formula for the sum of the interior angles in a polygon, with n = 5.

Sum of the interior angles = 180 × (5 - 2) = 540°

The angles must all add up to 540°.
Use this to form an equation in terms of x.

open parentheses 3 x space plus space 5 close parentheses plus x plus open parentheses x plus 40 close parentheses plus 95 plus open parentheses 2 x minus 20 close parentheses space equals space 540

Simplify by collecting the like terms.

table row cell 3 x space plus space x space plus space x space plus space 2 x space plus space 5 space plus space 40 space plus space 95 space minus space 20 space end cell equals cell space 540 end cell row cell 7 x space plus space 120 space end cell equals cell space 540 end cell end table

Solve the equation.

table row cell 7 x space plus space 120 space end cell equals cell space 540 end cell row cell 7 x space end cell equals cell space 420 end cell row cell x space end cell equals cell fraction numerator space 420 over denominator 7 end fraction end cell end table

bold italic x bold space bold equals bold space bold 60 bold degree

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.