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Angles in Polygons (OCR GCSE Maths)
Revision Note
Angles in Polygons
What is a polygon?
- 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
- For example:
- 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
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
- SUM OF INTERIOR ANGLES = 180° × (n – 2)
- 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°
- TOTAL OF EXTERIOR ANGLES = 360°
Examiner 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 .
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 .
Simplify by collecting the like terms.
Solve the equation.
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