National 5MathsSQARevision NotesGeometric SkillsProperties of Shapes & AnglesPolygons

Polygons (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Finding 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

    • A hexagon is a polygon with 6 sides

    • A heptagon is a polygon with 7 sides

    • A octagon is a polygon with 8 sides

    • A nonagon is a polygon with 9 sides

    • A decagon is a polygon with 10 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

A grid with eight geometric shapes in varying colours, each labelled: triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon.

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 an n-sided polygon can be split into n minus 2 triangles

  • It is best to remember the sums for the first few 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 exterior or interior angle in a regular polygon?

  • 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

    • This can be represented by the formula 360 over n

  • To find the size of an interior angle in a regular polygon:

    • Method 1
      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

    • Method 2
      Use the formula fraction numerator 180 open parentheses n minus 2 close parentheses over denominator n end fraction which calculates the interior angle directly

    • Method 3
      Find the size of the exterior angle and subtract it from 180°

      • For a pentagon, exterior angle is fraction numerator 360 degree over denominator 5 end fraction equals 72 degree

      • So interior angle is 180 minus 72 equals 108 degree

      • This method has less working than Method 1, and doesn't require you to remember a formula like in Method 2

  • 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?

  • To find a missing angle in a polygon:

    • Use the formula 180 degree cross times open parentheses n minus 2 close parentheses to work out the sum of the interior angles

    • Subtract the other interior angles in the polygon

How do I find the number of sides in a regular polygon?

  • If you are given the interior angle of a regular polygon

    • Method 1

      • set the angle equal to fraction numerator 180 open parentheses n minus 2 close parentheses over denominator n end fraction

      • then solve the equation to find n

    • Method 2

      • subtract the interior angle from 180° to find the exterior angle

      • set the exterior angle equal to 360 over n

      • then solve that equation to find n

Examiner Tips and Tricks

Make sure you identify whether you are dealing with a regular or irregular polygon before you start a question.

Even when a question asks about interior angles, it can often be easier

  • to work with exterior angles

  • then use interior angle + exterior angle = 180° to convert to an interior angle at the end

Worked Example

The interior angle of a regular polygon is 135°.

Determine the number of sides of the polygon.

Answer:

Method 1

Set the interior angle equal to fraction numerator 180 open parentheses n minus 2 close parentheses over denominator n end fraction

fraction numerator 180 open parentheses n minus 2 close parentheses over denominator n end fraction equals 135

Solve to find the value of n

table row cell 180 open parentheses n minus 2 close parentheses end cell equals cell 135 n end cell row cell 180 n minus 360 end cell equals cell 135 n end cell row cell 180 n minus 135 n end cell equals 360 row cell 45 n end cell equals 360 row n equals cell 360 over 45 end cell row n equals 8 end table

8 sides

Method 2

Use interior angle + exterior angle = 180° to find the exterior angle

exterior space angle equals 180 minus 135 equals 45

Set the exterior angle equal to 360 over n

45 equals 360 over n

Solve to find the value of n

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

8 sides

Worked Example

In the diagram shown below, ABCDEF is a regular hexagon.

  • Angle EGF is 31°.

  • AFG is a straight line.

A hexagon labelled A to F with a triangle extending from points E and F to point G. The angle at G measures 31 degrees. Dashed lines inside hexagon connecting centre to vertices.

Calculate the size of angle FEG.

Answer:

If you knew angle EFG, then you could use 180° in a triangle to find angle FEG

But angle EFG is the external angle of a regular hexagon (6 sides)

  • So use exterior space angle equals fraction numerator 360 degree over denominator n end fraction to find angle EFG

angle space EFG equals 360 over 6 equals 60

Therefore angle FEG is

180 minus 60 minus 31 equals 89

Angle FEG = 89°

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.