Angles in Polygons
What is a polygon?
- A polygon is a 2D shape with 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
What is the sum of the interior angles in a polygon?
- 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 -sided polygons can be split into triangles
- Sum of interior angles =
- 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:
- Divide by the number of sides ()
- For a pentagon:
- Find the sum of the interior angles
- To find the size of an exterior angle in a regular polygon:
- Divide 360° by the number of sides ()
- For a pentagon:
- Divide 360° by the number of sides ()
- 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
- 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°
- Regular hexagons tessellate
- 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°
- A regular hexagon will tessellate with two squares and an equilateral triangle
Regular 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 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
Substitute the 45 for the exterior angle
Solve by rearranging
Write down the name of a shape with 8 sides
Regular Octagon