How this instrument works
Every simple polygon's interior angles add up to (n − 2) × 180°, a fact that falls out of triangulation: pick one vertex, draw a diagonal to each vertex that isn't already adjacent to it, and the shape splits into exactly n − 2 non-overlapping triangles, each contributing its own 180°. A regular polygon simply divides that total evenly among its n identical corners, giving the interior angle (n − 2) × 180° ⁄ n. The exterior angle is what's left over on the same straight line — since the two are supplementary, exterior = 180° − interior collapses algebraically to the tidier 360° ⁄ n.
That 360° ⁄ n form has a reason independent of any single polygon: walk once around the boundary of any convex polygon, tracking only how much you turn at each corner, and the turns always add up to one full rotation, 360°, no matter how many sides the shape has or how long they are. A regular polygon just splits that fixed 360° into n identical turns, which is exactly the exterior angle. A turtle-graphics program leans on this directly to draw one: turn 360° ⁄ n after every side and the pen closes the shape on its own, without ever computing the interior angle.
Two edge cases bookend the formula. At the minimum, n = 3 gives an equilateral triangle with 60° interior corners and 120° exterior ones — the smallest simple polygon there is, since two sides alone can't enclose any area. At n = 4, the square is the one regular polygon where interior and exterior angles coincide exactly, both landing on 90°. Push n upward instead and the interior angle creeps toward 180° while the exterior angle shrinks toward 0°, the algebraic signature of a many-sided polygon flattening into something that looks, at a glance, like a circle — the same idea Archimedes used over two thousand years ago to bound the value of π by fencing a circle between inscribed and circumscribed polygons of ever more sides.
- Enter the side count into Number of sides, n — any whole number from 3 upward describes a valid simple polygon.
- Read Interior angle (regular polygon) for the angle at each vertex, shown in degrees by default, with radians and turns selectable.
- Read Exterior angle (regular polygon) for the supplementary angle formed by extending one side past the next vertex.
- Check the pair against each other: Interior angle plus Exterior angle always equals exactly 180°, whatever n you entered.
Worked example — a regular hexagon, n = 6
Set Number of sides, n to 6 for a regular hexagon — a stop sign's cousin, six equal sides and six equal corners. Interior angle (regular polygon) returns (6 − 2) × 180° ⁄ 6 = 720° ⁄ 6 = 120° exactly, stored internally as 2.0943951023931953 radians. Exterior angle (regular polygon) returns 360° ⁄ 6 = 60° exactly, or 1.0471975511965976 radians — and 120° + 60° = 180°, confirming the pair is supplementary as it must be for any n.
The same hexagon checks out two more ways. Its interior angles sum to (6 − 2) × 180° = 720° in total, and six corners at 120° apiece also add to 720°, so the whole-shape total and the single-vertex answer agree exactly. Walk the six exterior turns instead — six corners of 60° each — and they add to 360°, one complete rotation, the same full turn every convex polygon makes by the time its boundary closes back on itself.
Questions
What is the formula for the interior angle of a regular polygon?
Interior angle = (n − 2) × 180° ⁄ n, where n is the number of sides. The (n − 2) term comes from splitting the polygon into (n − 2) triangles by diagonals from one vertex, each contributing 180°; dividing that sum by n spreads it evenly across the regular polygon's equal corners. For n = 6 this gives exactly 120°.
How is the exterior angle related to the interior angle?
They're supplementary — interior plus exterior equals 180° at every vertex, because the exterior angle sits on the straight line formed by extending a side. Substituting the interior formula and simplifying gives the shortcut exterior = 360° ⁄ n directly, without ever needing the interior angle first.
Why do exterior angles always add up to 360°, no matter how many sides?
Because tracing the boundary of any convex polygon once around is exactly one full rotation, 360°, regardless of how many corners it takes to get there. A regular polygon just divides that fixed 360° into n equal turns, so each exterior angle is 360° ⁄ n — a triangle takes three big turns of 120°, a dodecagon takes twelve small turns of 30°, and both close the same single loop.
What's the smallest number of sides this formula works for?
Three. A triangle (n = 3) is the smallest simple polygon that can enclose an area, giving 60° interior and 120° exterior angles for the equilateral case. There is no polygon with fewer than three sides, so n = 2 or lower has no matching shape for the formula to describe.
Is there a regular polygon where interior and exterior angles are equal?
Yes — the square, n = 4. Both angles come out to exactly 90°, since (4 − 2) × 180° ⁄ 4 = 90° and 360° ⁄ 4 = 90° land on the same number. Every regular polygon with more sides has an interior angle larger than its exterior angle, and the triangle, with fewer, has it the other way around.
What happens to the angles as the number of sides grows very large?
The interior angle creeps toward 180° and the exterior angle shrinks toward 0°, since each is a smooth function of 1 ⁄ n. A 100-sided polygon already has a 176.4° interior angle and a 3.6° exterior one — visually close to a circle, the same many-sided-polygon idea Archimedes used to bound the value of π.