Find the exterior angle of a regular polygon from its sides, or calculate the number of sides from a known exterior angle. Also get the interior angle, angle sum, and polygon name.
Assumptions & Limitations
This exterior angle calculator helps you find the exterior angle of a regular polygon from its number of sides, or find the number of sides from a known exterior angle. It is designed for users who need a fast regular polygon exterior angle calculator, polygon angle converter, or exterior angle to sides calculator without working through the formulas manually.
The tool supports the two main regular polygon calculations users usually search for. First, it can calculate the exterior angle when the number of sides is known. Second, it can calculate how many sides a regular polygon has when the exterior angle is known. Along with the main answer, the calculator also returns the interior angle, the sum of exterior angles, and the polygon name when the result matches a valid regular polygon.
Because the tool works only with regular polygons, it assumes that all sides are equal and all interior angles are equal. That keeps the calculation exact and makes the results useful for common geometry problems involving polygons such as triangles, pentagons, hexagons, octagons, decagons, and other regular $n$-gons.
What This Exterior Angle Calculator Calculates
This calculator supports two exact workflows.
| Calculation mode | Input | Main result | Additional outputs |
|---|---|---|---|
| Find exterior angle from sides | Number of sides ($n$) | Exterior angle | Interior angle, sum of exterior angles, polygon name |
| Find sides from exterior angle | Exterior angle | Number of sides | Interior angle, sum of exterior angles, polygon name |
If you already know how many sides your shape has, entering the side count instantly provides the exact exterior angle required to close the shape. Conversely, if you measure or are given an exterior angle, entering that angle allows the tool to determine exactly how many sides the polygon must have. Both workflows automatically generate the corresponding interior angle, confirm the constant sum of the exterior angles, and return the formal geometric name of the resulting shape.
Unlike broad polygon tools, this calculator does not try to calculate area, perimeter, apothem, circumradius, or side length. It is focused only on regular polygon angle conversion and related outputs.
Exterior Angle Formula for a Regular Polygon
The main formula for a regular polygon exterior angle is:$$\text{Exterior angle} = \frac{360^\circ}{n}$$
Where $n$ is the number of sides.
If the exterior angle is already known, the number of sides is found by rearranging the formula:$$n = \frac{360^\circ}{\text{Exterior angle}}$$
The calculator also returns the interior angle using:$$\text{Interior angle} = 180^\circ – \text{Exterior angle}$$
And it shows the total sum of exterior angles:$$\text{Sum of exterior angles} = 360^\circ$$
These formulas apply to regular polygons only. If the reverse calculation gives a non-integer value for $n$, the result does not represent a real regular polygon in this calculator.
How to Use the Exterior Angle Calculator
Using the calculator is straightforward because it adapts to whichever starting value you have.
| Step | What to do |
|---|---|
| 1 | Choose whether you want to calculate from number of sides or from exterior angle |
| 2 | Enter the known value |
| 3 | Read the primary result shown by the calculator |
| 4 | Check the interior angle and sum of exterior angles |
| 5 | Review the polygon name if the input matches a valid regular polygon |
Start with the number of sides input when you are analyzing a known shape, such as a hexagon or a $15$-gon, and need its specific angle properties. Start with the exterior angle input when you are trying to identify an unknown regular polygon based on a single turn measurement.
Once you enter your value, the calculator processes the logic immediately. The returned interior angle tells you the inside measure at each vertex, providing the exact supplementary angle to your exterior result. The polygon name gives you the formal classification of the shape. If you use the reverse mode and your entered angle results in a decimal instead of a whole number of sides, the calculator will indicate that the input does not form a valid regular polygon.
Exterior Angle and Interior Angle Relationship
For every regular polygon, the exterior angle and interior angle at a vertex are supplementary. That means they always add up to $180^\circ$.$$\text{Interior angle} + \text{Exterior angle} = 180^\circ$$
That relationship is useful because once one angle is known, the other is immediate.
| Known value | Formula | Returned value |
|---|---|---|
| Exterior angle | $180^\circ – \text{Exterior angle}$ | Interior angle |
| Number of sides | $\frac{360^\circ}{n}$ | Exterior angle |
| Exterior angle | $\frac{360^\circ}{\text{Exterior angle}}$ | Number of sides |
This is why the calculator returns both angle values together. When a user calculates a polygon exterior angle, they often also need the corresponding interior angle for geometry homework, construction layout, or shape classification.
Common Regular Polygon Exterior Angles Table
A quick reference table helps cover the most common regular polygons and makes the page more useful than a formula-only tool.
| Polygon | Sides ($n$) | Exterior angle | Interior angle |
|---|---|---|---|
| Triangle | $3$ | $120^\circ$ | $60^\circ$ |
| Quadrilateral / Square | $4$ | $90^\circ$ | $90^\circ$ |
| Pentagon | $5$ | $72^\circ$ | $108^\circ$ |
| Hexagon | $6$ | $60^\circ$ | $120^\circ$ |
| Heptagon | $7$ | $51.4286^\circ$ | $128.5714^\circ$ |
| Octagon | $8$ | $45^\circ$ | $135^\circ$ |
| Nonagon | $9$ | $40^\circ$ | $140^\circ$ |
| Decagon | $10$ | $36^\circ$ | $144^\circ$ |
| Hendecagon | $11$ | $32.7273^\circ$ | $147.2727^\circ$ |
| Dodecagon | $12$ | $30^\circ$ | $150^\circ$ |
| Tridecagon | $13$ | $27.6923^\circ$ | $152.3077^\circ$ |
| Tetradecagon | $14$ | $25.7143^\circ$ | $154.2857^\circ$ |
| Pentadecagon | $15$ | $24^\circ$ | $156^\circ$ |
| Hexadecagon | $16$ | $22.5^\circ$ | $157.5^\circ$ |
| Heptadecagon | $17$ | $21.1765^\circ$ | $158.8235^\circ$ |
| Octadecagon | $18$ | $20^\circ$ | $160^\circ$ |
| Enneadecagon | $19$ | $18.9474^\circ$ | $161.0526^\circ$ |
| Icosagon | $20$ | $18^\circ$ | $162^\circ$ |
As the table shows, a smaller exterior angle means the polygon has more sides, while a larger exterior angle means it has fewer sides. As the side count increases, the exterior turn required at each vertex steadily decreases.
| Number of sides | Exterior angle | Quick interpretation |
|---|---|---|
| $3$ | $120^\circ$ | Few sides, large turn at each vertex |
| $4$ | $90^\circ$ | Quarter-turn at each vertex |
| $6$ | $60^\circ$ | Moderate turn, common regular polygon |
| $12$ | $30^\circ$ | Smaller turn, more sides |
| $20$ | $18^\circ$ | Many sides, small exterior turn |
Find the Number of Sides From an Exterior Angle
This calculator also works in reverse. If the exterior angle is known, divide $360^\circ$ by that value to find the number of sides.$$n = \frac{360^\circ}{\text{Exterior angle}}$$
This reverse lookup calculation works because traveling around the perimeter of any convex polygon always requires exactly one full $360^\circ$ rotation. By dividing that total $360^\circ$ by the size of a single exterior angle, you calculate exactly how many equal turns are needed to close the shape. Because a polygon must have complete, unbroken straight sides, this division must result in a whole number.
Here is how different angle inputs perform in the reverse calculation, mapping common angles to valid polygons:
| Exterior angle | Number of sides | Polygon |
|---|---|---|
| $120^\circ$ | $3$ | Triangle |
| $90^\circ$ | $4$ | Quadrilateral |
| $72^\circ$ | $5$ | Pentagon |
| $60^\circ$ | $6$ | Hexagon |
| $45^\circ$ | $8$ | Octagon |
| $40^\circ$ | $9$ | Nonagon |
| $36^\circ$ | $10$ | Decagon |
| $30^\circ$ | $12$ | Dodecagon |
| $24^\circ$ | $15$ | Pentadecagon |
| $18^\circ$ | $20$ | Icosagon |
If $360^\circ \div \text{Exterior angle}$ is not a whole number, the tool rejects the result.
| Exterior angle | Result | Valid regular polygon? |
|---|---|---|
| $72^\circ$ | $5$ sides | Yes |
| $45^\circ$ | $8$ sides | Yes |
| $50^\circ$ | $7.2$ sides | No |
| $22.5^\circ$ | $16$ sides | Yes |
Input Rules and Validation
The calculator includes simple but important validation rules so that the outputs stay mathematically valid.
| Input | Rule |
|---|---|
| Number of sides | Must be a whole number |
| Minimum sides | $3$ |
| Maximum sides | $1,000,000$ |
| Exterior angle | Must be greater than $0^\circ$ and less than $180^\circ$ |
| Reverse result | Must produce a whole number of sides |
These checks matter because not every number entered into the reverse formula represents a valid regular polygon. The validation keeps the tool focused on real regular polygon results only.
Example Calculations
Worked examples make the tool more useful for both quick checking and search coverage, demonstrating valid calculations alongside those that fail the integer validation rule.
| Input type | Input | Result |
|---|---|---|
| Sides | $3$ | Exterior angle = $120^\circ$, interior angle = $60^\circ$, sum of exterior angles = $360^\circ$, polygon = triangle |
| Sides | $6$ | Exterior angle = $60^\circ$, interior angle = $120^\circ$, sum of exterior angles = $360^\circ$, polygon = hexagon |
| Sides | $8$ | Exterior angle = $45^\circ$, interior angle = $135^\circ$, sum of exterior angles = $360^\circ$, polygon = octagon |
| Sides | $10$ | Exterior angle = $36^\circ$, interior angle = $144^\circ$, sum of exterior angles = $360^\circ$, polygon = decagon |
| Sides | $20$ | Exterior angle = $18^\circ$, interior angle = $162^\circ$, sum of exterior angles = $360^\circ$, polygon = icosagon |
| Exterior angle | $72^\circ$ | Number of sides = $5$, interior angle = $108^\circ$, sum of exterior angles = $360^\circ$, polygon = pentagon |
| Exterior angle | $50^\circ$ | Number of sides = $7.2$ (Invalid), interior angle = N/A, sum of exterior angles = N/A, polygon = N/A |
| Exterior angle | $36^\circ$ | Number of sides = $10$, interior angle = $144^\circ$, sum of exterior angles = $360^\circ$, polygon = decagon |
| Exterior angle | $30^\circ$ | Number of sides = $12$, interior angle = $150^\circ$, sum of exterior angles = $360^\circ$, polygon = dodecagon |
| Exterior angle | $22.5^\circ$ | Number of sides = $16$, interior angle = $157.5^\circ$, sum of exterior angles = $360^\circ$, polygon = hexadecagon |
Polygon Names the Calculator Can Identify
The tool returns a polygon name when the result matches one of its mapped side counts.
| Sides | Polygon name |
|---|---|
| $3$ | Triangle |
| $4$ | Quadrilateral |
| $5$ | Pentagon |
| $6$ | Hexagon |
| $7$ | Heptagon |
| $8$ | Octagon |
| $9$ | Nonagon |
| $10$ | Decagon |
| $11$ | Hendecagon |
| $12$ | Dodecagon |
| $13$ | Tridecagon |
| $14$ | Tetradecagon |
| $15$ | Pentadecagon |
| $16$ | Hexadecagon |
| $17$ | Heptadecagon |
| $18$ | Octadecagon |
| $19$ | Enneadecagon |
| $20$ | Icosagon |
For larger valid side counts, the calculator returns the shape as an $n$-gon.
What the Sum of Exterior Angles Means
The calculator always shows the sum of exterior angles as $360^\circ$. This is not a variable output for regular polygons. It is a fundamental geometric property of the full turn around the shape. Every regular polygon divides a full $360^\circ$ turn into equal, discrete exterior angles.
This constant total is the mathematical basis for both of the calculator’s modes. Because the sum is always $360^\circ$, the relationship between the sides and angles is locked. You can always find the exterior angle using $\frac{360^\circ}{n}$, and you can always find the number of sides using $\frac{360^\circ}{\text{Exterior angle}}$. As long as the shape is a regular polygon, these conversions will always balance perfectly.
That makes the sum of exterior angles useful as a quick check:
- if the polygon is regular, each exterior angle is one equal share of $360^\circ$
- multiplying one exterior angle by the number of sides returns $360^\circ$
- dividing $360^\circ$ by one exterior angle returns the number of sides
Assumptions and Limitations
This exterior angle calculator is intentionally narrow so it can stay accurate and fast.
| Assumption or limit | Meaning |
|---|---|
| Regular polygons only | All sides and all angles must be equal |
| Whole-number side count required | Reverse calculation must return an integer |
| Exterior angle must be between $0^\circ$ and $180^\circ$ | Values outside this range do not form a valid regular polygon in this tool |
| Minimum side count is $3$ | Fewer than $3$ sides is not a polygon |
| Sum of exterior angles is fixed | The tool always shows $360^\circ$ |
The tool does not calculate area, perimeter, side length, apothem, or angles for irregular polygons. It is built specifically for regular polygon exterior-angle calculations.
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