To find the measure of each exterior angle of a polygon, start with one key fact: the exterior angles of any convex polygon add up to 360°. Which means if the polygon is regular, all exterior angles are equal, so you can divide 360° by the number of sides. That single relationship is the foundation for solving most exterior-angle problems, whether you are working with a triangle, pentagon, hexagon, or a polygon with dozens of sides.
People argue about this. Here's where I land on it.
What Is an Exterior Angle?
An exterior angle is formed when you extend one side of a polygon and measure the angle between that extension and the adjacent side. At each vertex of a convex polygon, the interior angle and the exterior angle sit on a straight line, so they are supplementary:
Interior angle + Exterior angle = 180°
Simply put, if you know the interior angle, you can always find the exterior angle by subtracting from 180°:
Exterior angle = 180° − Interior angle
As an example, if a regular polygon has an interior angle of 150°, then each exterior angle is:
180° − 150° = 30°
That relationship is useful, but the fastest method for regular polygons uses the exterior-angle sum directly.
The Main Formula for a Regular Polygon
A regular polygon has all sides equal and all angles equal. Because its exterior angles are also equal, and because their total is always 360°, the measure of each exterior angle is:
Each exterior angle = 360° ÷ n
where n is the number of sides.
This formula works for every regular polygon, from an equilateral triangle to a regular 100-gon.
Quick Examples
- Equilateral triangle (n = 3): 360° ÷ 3 = 120°
- Square (n = 4): 360° ÷ 4 = 90°
- Regular pentagon (n = 5): 360° ÷ 5 = 72°
- Regular hexagon (n = 6): 360° ÷ 6 = 60°
- Regular octagon (n = 8): 360° ÷ 8 = 45°
- Regular decagon (n = 10): 360° ÷ 10 = 36°
- Regular dodecagon (n = 12): 360° ÷ 12 = 30°
- Regular 20-gon (n = 20): 360° ÷ 20 = 18°
Notice the pattern: as the number of sides increases, each exterior angle becomes smaller. A polygon with many sides looks more and more like a circle, so its exterior angles become very small Surprisingly effective..
Step-by-Step Method
Use this process when a problem asks for the measure of each exterior angle Simple, but easy to overlook..
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Identify whether the polygon is regular.
If all sides and angles are equal, you can use the division formula. If not, you cannot assume every exterior angle is the same. -
Count the number of sides.
The variable n represents the number of sides. A pentagon has n = 5, a hexagon has n = 6, and so on. -
Apply the formula.
Divide 360° by n:
Each exterior angle = 360° ÷ n -
Check using the interior angle.
The interior angle of a regular polygon is:
Interior angle = 180° − exterior angle
You can also use the interior-angle formula:
Interior angle = [(n − 2) × 180°] ÷ n
Both methods should give compatible results That's the whole idea.. -
Label your answer in degrees.
Exterior angles are measured in degrees, so include the degree symbol in your final answer Simple, but easy to overlook..
Example 1: Regular Pentagon
A regular pentagon has 5 sides.
Each exterior angle = 360° ÷ 5 = 72°
To verify, find the interior angle:
Interior angle = 180° − 72° = 108°
Using the interior-angle formula:
[(5 − 2) × 180°] ÷ 5 = (3 × 180°) ÷ 5 = 540° ÷ 5 = 108°
The results match, so each exterior angle is 72°.
Example 2: Regular Hexagon
A regular hexagon has 6 sides.
Each exterior angle = 360° ÷ 6 = 60°
The interior angle is:
180° − 60° = 120°
This is why regular hexagons appear so often in tiling patterns: their interior angles fit neatly around a point.
Example 3: Finding the Number of Sides From an Exterior Angle
Sometimes you know the exterior angle and need to find how many sides the polygon has. Rearrange the formula:
n = 360° ÷ each exterior angle
If each exterior angle of a regular polygon is 24°, then:
**n =
Example 4: Finding the Number of Sides From an Exterior Angle
Sometimes you know the exterior angle and need to determine how many sides the polygon has. Rearranging the same relationship gives:
(n = \dfrac{360^\circ}{\text{ext exterior angle}})
For a regular polygon whose each exterior angle measures 24°:
[ n = \frac{360^\circ}{24^\circ} = 15 ]
Thus a regular polygon with an exterior angle of (24^\circ) must be a 15‑gon (a regular pentadecagon).
To double‑check, use the interior‑angle connection:
- Interior angle = (180^\circ - 24^\circ = 156^\circ).
- Using the interior‑angle formula: (\displaystyle \frac{(n-2)\times180^\circ}{n}= \frac{(15-2)\times180^\circ}{15}= \frac{13\times180^\circ}{15}= \frac{2340^\circ}{15}=156^\circ), which matches.
The results are consistent, confirming that (n = 15) Nothing fancy..
Conclusion
When dealing with regular polygons, the size of a single exterior angle tells us much about the shape’s structure. By dividing the full circle’s measure—(360^\circ)—by the number of equal exterior angles, we obtain both the individual exterior angle for a given side count and the total number of sides if the exterior angle is known. As the number of sides grows, each exterior angle shrinks, making the polygon appear more circular. This simple division method unites geometry, symmetry, and the idea of approaching a perfect circle through increasingly complex polygons Worth keeping that in mind..
