Circle Theorems Calculator (inscribed, central, tangent & chord angles)

Understanding the circle theorems calculator — concept and principle

This calculator applies the standard circle theorems: inscribed and central angles, opposite angles of a cyclic quadrilateral, the tangent–chord angle, angles formed by chords crossing inside or secants and tangents meeting outside, and the intersecting-chord and tangent–secant length rules. Choose a theorem, enter the known values, and it gives the missing angle, arc or length with the rule used.

Both angles stand on the same arc; the one at the edge is half the one at the centre.

How it is calculated

Inscribed angle = central angle ÷ 2 (and central = 2 × inscribed) on the same arc.
Cyclic quadrilateral: opposite angles add up to 180°.
Tangent–chord angle = intercepted arc ÷ 2.
Two chords crossing inside: angle = (arc₁ + arc₂) ÷ 2.
Secants or tangents meeting outside: angle = (far arc − near arc) ÷ 2.
Intersecting chords: a × b = c × d. Tangent–secant: t² = outside part × whole secant.

Angles and arcs are in degrees; lengths in any unit.

The principle

The key result is the inscribed angle theorem. Draw radii from the centre to the ends of the arc and to the point on the circle: this makes isosceles triangles, and adding their base angles shows that the angle at the circle is half the angle at the centre. Every other angle rule follows from it: an angle in a semicircle stands on a 180° arc, so it is 90°; the two opposite angles of a cyclic quadrilateral stand on arcs that together make 360°, so they add up to 180°. The length rules come from similar triangles formed by the chords or secants.

Worked example

Default inputs: central angle → inscribed angle, value 1 = 110°.

Rule used: inscribed = central ÷ 2.
Inscribed angle = 110° ÷ 2 = 55°.
What was found: Inscribed angle (°). (Values 2 and 3 are not used by this theorem.)

Each point is the calculator re-run with only “Value 1 (angle, arc or length)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Value 1 (angle, arc or length)	Result
27.5	13.75
55	27.5
82.5	41.25
110	55
138	68.75
165	82.5
220	110

How to read the result

Check that the answer fits the picture: an inscribed angle is always less than 180°, and an angle standing on a major arc is obtuse. If you know an angle and need an arc, run the theorem backwards: the arc in degrees equals the central angle. For length rules, the products must balance, which is a quick check on any measurement.

Common mistakes

Using the wrong arc, such as the major arc instead of the minor arc the angle stands on.
Adding the arcs for an angle outside the circle; outside angles use the difference.
Forgetting to halve the arc for tangent–chord angles.
Using the inside segment instead of the whole secant in t² = outside × whole.

Keep in mind: The theorems assume a true circle and points exactly on it. Measured diagrams have errors, so results are approximate.

Frequently asked questions

What is the inscribed angle theorem?

An angle at the circle is half the central angle standing on the same arc.

Why is the angle in a semicircle 90°?

It stands on a 180° arc, and half of 180° is 90°.

What is the intersecting chords theorem?

When two chords cross, the products of their parts are equal: a × b = c × d.

Sources: Euclid, Elements, Book III (circle theorems)

Checked 9 Nov 2026. Standard theorems of circle geometry (inscribed-angle theorem, cyclic quadrilateral, tangent–chord angle, angles formed by chords and by secants or tangents, intersecting-chords and tangent–secant products), as in Euclid’s Elements Book III and any school geometry course.

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