Angles in Polygons

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

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

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

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 $x$ .

$(3 x + 5) + x + (x + 40) + 95 + (2 x - 20) = 540$

Simplify by collecting the like terms.

$\begin{array}{rcl} 3 x + x + x + 2 x + 5 + 40 + 95 - 20 & = & 540 \\ 7 x + 120 & = & 540 \end{array}$

Solve the equation.

$\begin{array}{rcl} 7 x + 120 & = & 540 \\ 7 x & = & 420 \\ x & = & \frac{420}{7} \end{array}$

$x = 60 °$

Angles in Polygons (CIE IGCSE Maths: Extended)