The measure of one exterior angle of a regular polygon is found by dividing 360 degrees by the number of sides the polygon has.
Understanding Exterior Angles in Polygons
An exterior angle of a polygon is formed by extending one of its sides and measuring the angle between the extended side and the adjacent side. In any convex polygon, regardless of whether it's regular or irregular, the sum of all its exterior angles always adds up to 360 degrees. This fundamental property holds true for shapes ranging from triangles to decagons and beyond.
Calculating One Exterior Angle in a Regular Polygon
For a regular polygon, all its sides are of equal length, and all its interior angles (and consequently, all its exterior angles) are equal. This uniformity simplifies the calculation significantly.
To determine the measure of a single exterior angle in a regular polygon, use the following formula:
Exterior Angle = 360° / Number of Sides (n)
Where:
360°is the total sum of all exterior angles for any polygon.nrepresents the number of sides (or vertices) of the regular polygon.
This formula works because, with all exterior angles being equal in a regular polygon, dividing their total sum by the count of angles (which is equal to the number of sides) gives you the measure of one individual angle.
Examples of Exterior Angles in Regular Polygons
Let's look at how this applies to common regular polygons:
| Regular Polygon | Number of Sides (n) | Calculation (360° / n) | Measure of One Exterior Angle |
|---|---|---|---|
| Equilateral Triangle | 3 | 360° / 3 | 120° |
| Square | 4 | 360° / 4 | 90° |
| Regular Pentagon | 5 | 360° / 5 | 72° |
| Regular Hexagon | 6 | 360° / 6 | 60° |
| Regular Octagon | 8 | 360° / 8 | 45° |
| Regular Decagon | 10 | 360° / 10 | 36° |
Practical Insights and Solutions
- Equilateral Triangle: A triangle with 3 equal sides will have three exterior angles, each measuring 120° (360° / 3).
- Square: A square has 4 equal sides, resulting in four exterior angles, each 90° (360° / 4). This aligns with its interior angles also being 90°.
- Regular Pentagon: With 5 sides, each exterior angle of a regular pentagon is 72° (360° / 5).
- Regular Hexagon: A six-sided regular polygon features exterior angles of 60° each (360° / 6).
Understanding exterior angles is crucial in various geometric applications, from architecture and design to physics and engineering. For more comprehensive information on polygon properties, you can explore resources like Khan Academy's geometry section.
