Angles in Polygons

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°

Exam 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 $x$ .

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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