Circle Theorems Calculator

Background.

Draw a chord across a circle and pick a point on the circle on one side of it. Join that point to the chord's two ends, and the angle you make is the same wherever on that side you put the point. Slide it along and the triangle changes shape completely, yet the angle at the circle never moves. That constancy is the inscribed angle theorem, and the angle's size is fixed by one thing: the arc on the far side, which the angle is said to intercept. The angle is exactly half of that arc.

Most of the circle theorems in school geometry are this one rule in different clothes. An angle in a semicircle intercepts 180° of arc, so it must be 90°. Two angles standing on the same arc are equal because each is half of it. Opposite corners of a quadrilateral drawn inside a circle add to 180° because between them they intercept all 360°. And the angle at the centre is twice the angle at the circumference because a central angle simply equals its arc.

Give the page the intercepted arc in degrees and it returns the inscribed angle. In a real problem the work is usually in finding the right arc. It is the one lying inside the angle, across the circle from the vertex, and when the vertex sits on the shorter arc the intercepted arc is the longer one, so the angle comes out obtuse.

Arc measure means degrees of turn, the same as the central angle, not the length of the curve. The angle between a tangent and a chord obeys the same halving rule as a limiting case. Angles made by chords crossing inside the circle, or by secants meeting outside it, combine two arcs instead, and those are not computed here.

What is circle theorems calculator?

An inscribed angle has its vertex on a circle and its two sides running along chords. The arc it intercepts is the part of the circle between the two chord ends that lies inside the angle. Arc measure is given in degrees and equals the central angle standing on the same arc, so a full circle measures 360°. The inscribed angle theorem says the inscribed angle is exactly half its intercepted arc, whatever the size of the circle and wherever on the remaining arc the vertex sits.

How to use this calculator.

1. 01Find the arc the angle intercepts by tracing from one chord end to the other along the side of the circle away from the vertex. If the problem gives the central angle on that arc instead, use it; the number is the same.
2. 02If you only know the arc the vertex sits on, subtract it from 360° first. A vertex on a 100° arc intercepts the remaining 260°.
3. 03If you have an arc length and a radius rather than degrees, convert first: arc in degrees = arc length ÷ radius × 180 ÷ π.
4. 04Enter the arc, from 0° to 360°, and read the inscribed angle in degrees. The two ends of that range are limiting cases where the angle collapses, not angles you can actually draw.

The formula.

Inscribed angle = intercepted arc ÷ 2. The proof is short. Join the centre O to the vertex P. When one side of the angle, PA, passes through the centre, the other side PB and the radius OB form triangle OPB, which is isosceles because OP and OB are both radii. Its two base angles are equal, and the exterior angle at O, which is the central angle on the arc, equals their sum: twice the inscribed angle. Every other position of P splits into two such cases, added or subtracted. Since an arc's measure equals its central angle, halving the arc gives the inscribed angle. Nothing else enters: not the radius, not the chord length, not where the vertex sits, provided it stays off the intercepted arc. An arc of 180° gives a right angle, smaller arcs give acute angles and larger arcs obtuse ones.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.