Why is the exterior-angle total 360°, not the interior total?

Exterior turns track the change in heading around one closed convex boundary. Interior sums track the inside corners and grow by 180° per added side. They count different geometric quantities.

Does a 45° exterior turn identify a regular octagon?

Under the convex regular assumption, 360/45 gives eight sides. The angle alone does not verify that a real eight-sided object has equal sides or equal angles.

Can one exterior angle be 150° in this tool?

A convex regular polygon has at least three sides, giving a maximum equal turn of 120°. Individual turns in some irregular shapes can be larger, but they do not satisfy this tool's regular-polygon assumption.

Exterior Angle Calculator

The exterior angle calculator finds the equal turning angle at each vertex of a convex regular polygon, or tests the number of sides implied by a known exterior angle. The diagram extends the incoming side beyond a vertex and marks the angle between that extension and the next side. This identifies the exterior region directly, which helps distinguish a boundary turn from the inside corner or an angle drawn at the polygon's center.

Follow a boundary turn with the exterior angle calculator

Imagine walking along the sides in one consistent direction. At each corner, change heading to follow the next side. After one circuit of a convex polygon, the total turning is 360°. For a regular shape the turns are equal, so one exterior angle is the full turn divided by the number of sides. These are the ordinary adjacent exterior angles, not arbitrary outside regions or reflex angles around the vertex.

Start with a side count or a turning angle

Select Number of sides to calculate a regular exterior angle. Enter a whole number from 3 to 10,000. The result also shows the associated regular interior angle and interior-angle sum for comparison.
Select Exterior angle to infer a regular side count. Enter a degree value greater than zero and no greater than 120°. The largest supported ordinary exterior angle belongs to an equilateral triangle.
Read the equation 360° / n or 360° / exterior angle. In inverse mode, a value inside the broad angular range may still imply a fractional count. Such a result is rejected instead of being rounded into a polygon.
Compare the marked extension with the interior region. Their adjacent angles total 180°. The diagram shows an ideal regular outline through 24 sides; larger counts use a conceptual illustration while keeping the actual input in the equations.

Exterior-angle formulas and inverse tolerance

For a convex regular n-sided polygon, E = 360° / n. The total of one exterior turn at each vertex is 360°, and all turns are equal. Dividing that total is justified by regularity, not by the number of sides alone.
The related interior corner I = 180° − E because the side extension and its original side form a straight line. The inverse equation is n = 360° / E, followed by an integer and range check.
This tool shares the site's strict inverse helper. An estimated count must lie within 0.0000001 of an integer, and the angle reconstructed from that integer must agree within 0.0000001 degree. The accepted count is a numerical match under this tolerance.
A convex irregular polygon also completes a 360° exterior-turn total, but its individual turns can differ. For concave boundaries, signed turning conventions are needed; simple positive values at every corner cannot describe that situation correctly. This calculator's one-angle modes stay within the convex regular case.

Examples of regular turns

A regular hexagon has exterior angle 360° / 6 = 60° and interior angle 180° − 60° = 120°. Six equal turns complete the full circuit. The exterior angle is half the interior value in this example, but that proportion does not hold for every polygon.
A square has four 90° turns. Its exterior and interior angles happen to have equal sizes, while they occupy different adjacent regions. An equilateral triangle has a 120° exterior turn and a 60° interior corner.
An exterior angle of 24° gives n = 360° / 24° = 15, with interior angle 156°. An exterior angle of 50° gives 7.2 sides, which is not an exact regular polygon and receives a validation message.
In a drawing or programming lesson, students can use the turn value to repeat a straight segment and a heading change around a regular outline. A complete boundary needs n such turns, including the final turn returning to the starting heading. Omitting that last turn leads to an incorrect total.

Exterior and central angles share a value, not a location

For a regular polygon, the central angle between radii to adjacent vertices also equals 360° / n. That numerical equality follows from equal subdivisions of a full turn at the center. The exterior turn is located on the boundary and uses a side extension. They should not be labeled interchangeably on a diagram.

Use the broader polygon calculator to compare exterior and central values alongside names and diagonals. Use the interior calculator to focus on the sum of the inside corners and the assumptions behind a single regular corner. These are useful next steps when a problem changes what is known or which region is requested.

Small turns, approximate measurements, and diagram limits

The exterior angle calculator displays ordinary decimals to at most six places while retaining internal precision. Very small or large values use six significant digits. A measured approximate angle can imply a noninteger count; the tool cannot prove regularity from that measurement. Values below the angle of a 10,000-sided regular polygon require an unsupported count and are rejected. The conceptual diagram for larger counts is labeled and does not draw thousands of vertices. Do not confuse an exterior turn with the reflex region outside the polygon, and do not apply equal-turn division to an irregular shape unless equality is independently known.