Angles in polygons (Edexcel GCSE Maths): Revision Notes

Angles in polygons

Understanding interior and exterior angles

Interior angles are the angles inside a polygon, while exterior angles are formed when you extend one side of the polygon outward. These two types of angles are fundamental to solving polygon problems.

When you look at any vertex of a polygon, the interior and exterior angles always add up to 180° because they form a straight line. This relationship is crucial for solving many polygon questions.

Key formulas for any polygon

For any polygon with n sides, you can use these essential formulas:

• Sum of interior angles = $180° \times (n - 2)$
• Sum of exterior angles = $360°$ (this is always the same, regardless of the number of sides)

The interior angle formula works because you can divide any polygon into triangles. A polygon with n sides can be divided into (n - 2) triangles, and since each triangle has angles totalling 180°, you multiply by (n - 2).

The exterior angles always sum to 360° because if you walk around the outside of any polygon and return to your starting position, you will have turned through one complete rotation.

Regular polygons

A regular polygon has all sides equal in length and all angles equal in size. This makes calculations much simpler.

For a regular polygon with n sides:

• Each exterior angle = $\frac{360°}{n}$
• Each interior angle = $180° - \text{exterior angle}$

Examples of regular polygons:

• Regular pentagon (5 sides): each exterior angle = $\frac{360°}{5} = 72°$
• Regular hexagon (6 sides): each exterior angle = $\frac{360°}{6} = 60°$
• Regular octagon (8 sides): each exterior angle = $\frac{360°}{8} = 45°$

Problem-solving strategies

Working with exterior angles is often easier than working with interior angles. Here's why:

• Exterior angles are simpler to calculate (just divide 360° by the number of sides for regular polygons)
• You can use the fact that interior and exterior angles on a straight line add up to 180°
• Once you find the exterior angle, finding the interior angle is straightforward

Worked example approach

When solving polygon problems, follow these systematic steps:

1. Identify whether the polygon is regular (all angles equal) or irregular
2. Choose whether to work with interior or exterior angles based on which seems simpler
3. Apply the appropriate formula
4. Check your answer makes sense (angles should be reasonable sizes)

Exam tips

• Use exterior angles when possible - they're usually easier to work with
• Remember that exterior angles always sum to 360°, regardless of the polygon
• Check whether the polygon is regular before assuming all angles are equal
• Verify your answers make sense - interior angles in regular polygons should be reasonable sizes