Angles in Polygons (Edexcel GCSE Maths: Foundation)

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Angles in Polygons

What is a polygon?

  • A polygon is a 2D shape with n straight sides
    • 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 size
    • A regular polygon with 3 sides is an equilateral triangle
    • A regular polygon with 4 sides is a square

What are the interior angles and the exterior angles of a polygon?

  • Interior angles are the angles inside a polygon at the corners
  • The exterior angle at a corner is the angle needed to make a straight line with the interior angles
    • It is not the angle that forms a full turn at the corner
  • The interior angle and exterior angle add up to 180° at each corner

interior and exterior angles summing to 180 degrees

Interior and exterior angles in a hexagon

What is the sum of the interior angles in a polygon?

  • To find the sum of the interior angles in a polygon of n sides, use the rule
    • Sum of interior angles = 180 degree space cross times space left parenthesis n space – space 2 right parenthesis
      • This formula comes from the fact that n-sided polygons can be split into n minus 2 triangles
  • Remember the sums for these polygons
    • The interior angles of a triangle add up to 180°
    • The interior angles of a quadrilateral add up to 360°
    • The interior angles of a pentagon add up to 540°

What is the sum of the exterior angles in a polygon?

  • The exterior angles in any polygon always sum to 360°

How do I find the size of an interior or exterior angle in a regular polygon?

  • To find the size of an interior angle in a regular polygon:
    • Find the sum of the interior angles
      • For a pentagon: 180 degree cross times open parentheses 5 minus 2 close parentheses space equals space 540 degree
    • Divide by the number of sides (n)
      • For a pentagon: 540 degree divided by 5 equals 108 degree
  • To find the size of an exterior angle in a regular polygon:
    • Divide 360° by the number of sides (n)
      • For a pentagon: 360 degree divided by 5 equals 72 degree
  • The interior angle and exterior angle add to 180°
    • Subtract the exterior angle from 180° to find the interior angle
    • Subtract the interior angle from 180° to find the exterior angle
Regular Polygon Number of Sides Sum of Interior Angles Size of Interior Angle Size of Exterior Angle
Equilateral Triangle 3 180° 60° 120°
Square 4 360° 90° 90°
Regular Pentagon 5 540° 108° 72°
Regular Hexagon 6 720° 120° 60°
Regular Octagon 8 1080° 135° 45°
Regular Decagon 10 1440° 144° 36°

How do I find a missing angle in a polygon?

  • STEP 1
    Calculate the sum of the interior angles for the polygon
    • Use the formula 180 degree cross times open parentheses n minus 2 close parentheses
  • STEP 2
    Subtract the other interior angles in the polygon

What is tessellation?

  • Some shapes can arranged so that their corners fit together exactly, without gaps
    • This is called tessellation
  • This only works when the interior angles that are being put next to each other sum to 360°
    • Regular hexagons tessellate
      • The interior angle is 120° which is a factor of 360°
      • Therefore 3 hexagons will fit together with no gaps
    • Regular pentagons do not tessellate with other regular pentagons
      • The interior angle is 108° which is not a factor of 360°

Hexagons tessellate as the three angles that meet sum to 360 degrees

  • Combinations of different shapes can also tesselate
    • A regular hexagon will tessellate with two squares and an equilateral triangle
      • Their interior angles are 120°, 90°, and 60° respectively
      • 120 + (2×90) + 60 = 360°

Hexagon, square, and equilateral triangle tessellatingRegular hexagons, squares, and equilateral triangles tessellating

Examiner Tip

  • Make sure you identify whether you are dealing with a regular or irregular polygon before you start a question
  • Finding the sum of the interior angles using 180 cross times open parentheses n minus 2 close parentheses can often be a good starting point for finding missing angles

Worked example

The exterior angle of a regular polygon is 45°.

Write down the name of the polygon.

The formula for the exterior angle of a regular polygon is Exterior space Angle equals fraction numerator 360 degree over denominator n end fraction 

Substitute the 45 for the exterior angle

45 degree equals fraction numerator 360 degree over denominator n end fraction

Solve by rearranging

table row n equals cell 360 over 45 end cell row n equals 8 end table

Write down the name of a shape with 8 sides

Regular Octagon

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.