Inscribed angle calculator

Give it the arc measure in degrees and it returns the inscribed angle in degrees, plus the radian equivalent for free. The computation is a single halving — no iteration, no approximation — so the same input always yields the same answer, whether you are checking a geometry exercise, laying out a circular pattern, or validating angles in a drawing pipeline. Arc measures from 0 to 360 degrees are accepted; anything outside that range is rejected with a clear error.

The inscribed angle theorem, in one step

The inscribed angle theorem states that an angle inscribed in a circle is half the central angle that subtends the same arc — and since the central angle equals the arc measure, the inscribed angle is simply half the arc. A 90-degree arc gives a 45-degree inscribed angle; a semicircle of 180 degrees gives exactly 90 degrees, which is Thales' theorem in disguise: any angle inscribed in a semicircle is a right angle. That is why carpenters can test whether a corner is square by checking it against a circle drawn through its endpoints. The calculator performs that single halving and also converts the result to radians by multiplying by π over 180, because radians are the natural unit in trigonometry, physics and most programming libraries, while degrees are what textbooks, protractors and construction drawings speak. Because the relationship is a plain ratio, the result is exact and deterministic: there is no solver to converge and no tolerance to tune, so a browser widget and a paid API call return byte-identical answers for the same input.

Inputs, validation and edge cases

The calculator needs exactly one number: the measure of the intercepted arc in degrees. It must be finite and fall between 0 and 360 inclusive. An arc of 0 degrees is legitimate — a degenerate arc collapsed to a point — and yields an inscribed angle of 0. An arc of exactly 360 degrees, the full circle, is also accepted and yields 180 degrees. Anything negative, or anything above a full circle, is geometrically meaningless for an intercepted arc and is rejected with a clear validation error instead of being silently wrapped or clamped; in an automated pipeline that distinction matters, because a malformed input stops the run rather than contaminating downstream results. The field also answers to common aliases — arc, arc_degrees, intercepted_arc — so payloads from different sources integrate without remapping. String inputs like '120' are coerced when they parse as plain numbers. Outputs are rounded to twelve decimal places, which keeps the JSON stable across machines while staying far beyond any practical measurement precision.

Where people actually use it

Students meet this theorem in every circle-geometry unit: given an arc, find the angle, or spot that two inscribed angles standing on the same arc must be equal. This calculator is the quick check that confirms a hand derivation landed on the right value. Designers and makers use it when a circular pattern is specified by arcs — a decorative fan, a rose window, a gear tooth profile — and they need the vertex angles to cut or draw. Navigation and surveying inherit it through the horizontal-angle fix: a navigator who measures the angle between two landmarks sits on the arc of a circle through them, so the relationship runs both ways. In software it appears wherever circular geometry is reconstructed or verified: validating angles in a CAD export, generating polygon-on-circle artwork, or computing sight angles in a game. In every case the contract is the same — one arc measure in degrees in, one angle out — and that is exactly what this capability exposes. It runs free in your browser on this page, and the API charges $0.002 per request when you automate it.