Exterior Angle (Regular Polygon)

Statement

The exterior angle of a regular polygon is the angle formed between one side of the polygon and the extension of its neighbouring side. For any regular polygon (where all sides and angles are equal), the formula is:

\[\text{Exterior angle} = \frac{360^\circ}{n}\]

where \(n\) is the number of sides of the polygon.

Why it works

• The sum of all exterior angles of any polygon is always \(360^\circ\), no matter how many sides it has.
• In a regular polygon, all exterior angles are equal. So if there are \(n\) sides, then each exterior angle is \(360^\circ \div n\).
• This fact also helps calculate the interior angles, since interior and exterior angles are supplementary: \[\text{Interior angle} = 180^\circ - \text{Exterior angle}\].

Recipe (Steps to Apply)

1. Identify the number of sides \(n\).
2. Apply the formula \(\text{Exterior angle} = \tfrac{360^\circ}{n}\).
3. If required, calculate the interior angle using \(180^\circ - \text{Exterior angle}\).

Spotting it

Use this formula when a question involves regular polygons, and you are asked to find the size of an exterior angle, or to determine the number of sides from a given angle.

Common pairings

• Interior angle of a regular polygon.
• Number of sides of a polygon given its angles.
• Sum of interior angles: \((n-2)\times 180^\circ\).

Mini examples

1. Square: \(n=4\). Exterior angle = \(360^\circ / 4 = 90^\circ\).
2. Regular hexagon: \(n=6\). Exterior angle = \(360^\circ / 6 = 60^\circ\).

Pitfalls

• Forgetting that this formula only works for regular polygons (all sides equal).
• Mixing up exterior and interior angles.
• Forgetting that the sum of all exterior angles is always \(360^\circ\).

Exam Strategy

• If asked for number of sides, rearrange the formula: \(n = 360^\circ / \text{Exterior angle}\).
• Always label whether you are calculating interior or exterior angles.
• Check answers make sense (angles should be less than 180° for convex polygons).

Worked Micro Example

Find the exterior angle of a regular decagon (10 sides).

\(\tfrac{360^\circ}{10} = 36^\circ\).

So each exterior angle = \(36^\circ\).

Summary

The exterior angle of a regular polygon is found by dividing \(360^\circ\) by the number of sides. This result is fundamental in polygon geometry and is closely linked to interior angles and polygon classification.