Angles in Polygons

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

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

To find the sum of the interior angles in a polygon of sides, use the rule
- Sum of interior angles =
  - This formula comes from the fact that $n$ -sided polygons can be split into $n - 2$ 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°

To find the size of an interior angle in a regular polygon:
- Find the sum of the interior angles
  - For a pentagon: $180 ° \times (5 - 2) = 540 °$
- Divide by the number of sides ()
  - For a pentagon: $540 ° \div 5 = 108 °$
To find the size of an exterior angle in a regular polygon:
- Divide 360° by the number of sides ()
  - For a pentagon: $360 ° \div 5 = 72 °$
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

STEP 1
Calculate the sum of the interior angles for the polygon
- Use the formula $180 ° \times (n - 2)$
STEP 2
Subtract the other interior angles in the polygon

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°

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°

Regular hexagons, squares, and equilateral triangles tessellating

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 $180 \times (n - 2)$ can often be a good starting point for finding missing angles

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 $Exterior Angle = \frac{360 °}{n}$

Substitute the 45 for the exterior angle

$45 ° = \frac{360 °}{n}$

Solve by rearranging

$\begin{array}{rcl} n & = & \frac{360}{45} \\ n & = & 8 \end{array}$

Write down the name of a shape with 8 sides

Regular Octagon

Angles in Polygons (Edexcel GCSE Maths: Foundation)