Calculate an Exterior Angle
Choose a calculation method. Results appear above this form after submission.
Formula Used
Regular polygon: e = 360° ÷ n
Interior relationship: e = 180° − i
Missing exterior angle: emissing = 360° − Σeknown
Regular side count: n = 360° ÷ e
Here, e means an exterior angle, i means an interior angle, and n means the number of sides. These formulas use one exterior angle at each vertex of a convex polygon.
How to Use This Calculator
- Select the method matching your known information.
- Enter a side count, angle, or known exterior-angle sum.
- Choose degree or radian input and output units.
- Select the rounding precision that fits your task.
- Press the calculation button and review the result above.
- Download a CSV or PDF after a successful calculation.
Example Data Table
| Polygon | Sides | Each Exterior Angle | Each Interior Angle | Total Exterior Turn |
|---|---|---|---|---|
| Equilateral triangle | 3 | 120° | 60° | 360° |
| Square | 4 | 90° | 90° | 360° |
| Regular pentagon | 5 | 72° | 108° | 360° |
| Regular hexagon | 6 | 60° | 120° | 360° |
| Regular octagon | 8 | 45° | 135° | 360° |
| Regular decagon | 10 | 36° | 144° | 360° |
Exterior Angles in Geometry
Exterior angles describe the outside turns made while tracing a polygon. They support geometry, drafting, mapping, design, and programming. A convex polygon has one exterior angle at every vertex. Walk around the shape in one direction. Each turn contributes to a complete rotation.
The total of one exterior angle at each convex vertex is always 360 degrees. This rule works for regular and irregular convex polygons. When several exterior angles are known, subtract their sum from 360. The remainder is the missing exterior angle.
Regular Polygon Calculations
Regular polygons make the calculation even simpler. All exterior angles are equal. Divide 360 by the number of sides. Triangles have 120-degree exterior angles. Squares have 90-degree exterior angles. A regular pentagon has exterior angles of 72 degrees. More sides create smaller exterior angles.
Interior and Exterior Relationships
Interior and exterior angles at the same convex vertex are supplementary. Their sum equals 180 degrees. Subtract an interior angle from 180 to find its exterior angle. This relationship is especially helpful when a drawing labels only one angle type.
Finding the Side Count
Use the side count to identify a regular polygon from its exterior angle. Divide 360 by the exterior angle. A whole-number result gives the number of sides. A 45-degree exterior angle produces eight sides. A non-whole result means the angle does not describe an exact regular polygon.
Choosing the Best Method
This calculator supports several methods. Choose regular polygon mode when the number of sides is known. Choose interior angle mode when one interior measure is given. Choose exterior angle mode to estimate the side count. Choose missing angle mode for irregular convex polygons. Enter angles in degrees or radians. Select your preferred output unit and rounding precision.
Checking Your Work
Check that every entered value is sensible. Convex interior angles must stay between zero and 180 degrees. Convex exterior angles must be greater than zero and no more than 180 degrees. Regular polygons need at least three sides. The calculator highlights invalid entries before showing a result.
Saving the Result
Use the result table as a study record or design reference. Download a CSV file for spreadsheets. Export a PDF copy for printing or sharing. Compare your values with the example table below. Recheck the formula and units before using a result in a drawing. Accurate inputs lead to dependable geometry decisions. This makes checks easier during homework, layout, and planning.
Frequently Asked Questions
1. What is an exterior angle?
It is the angle formed outside a polygon when one side is extended. It represents the turning amount at a vertex.
2. Why do exterior angles total 360 degrees?
Walking once around a convex polygon returns you to the starting direction. That complete turn equals 360 degrees.
3. How do I find each exterior angle of a regular polygon?
Divide 360 degrees by the number of sides. Every exterior angle is equal in a regular polygon.
4. What is the relationship between interior and exterior angles?
At one convex vertex, an interior angle and its adjacent exterior angle add to 180 degrees.
5. Can I use radians?
Yes. Enter radians as the input unit. You can also choose radians for the displayed result.
6. Can this calculate an irregular polygon?
Yes. Use missing exterior angle mode when all but one exterior angle are known. Their total remains 360 degrees for convex polygons.
7. What does a non-whole side count mean?
It means the entered exterior angle does not create an exact regular polygon. The angle may still be valid for another shape.
8. Why must a regular polygon have at least three sides?
A polygon is a closed figure made from at least three line segments. Two segments cannot form a polygon.
9. Does this work for concave polygons?
The calculator uses the standard positive-turn convention for convex polygons. Concave polygons require careful direction and signed-angle handling.
10. What precision should I choose?
Use zero decimals for simple classroom answers. Use two to four decimals for measurements, layouts, or technical estimates.
11. What is included in the downloaded files?
Both exports include the displayed calculation rows. CSV works with spreadsheet programs, while PDF gives a compact printable record.