Inscribed Quadrilateral Calculator

An inscribed quadrilateral (also called a cyclic quadrilateral) is a quadrilateral whose four vertices all lie on a single circle. Its most-useful property is the opposite angles are supplementary rule:

∠A + ∠C = 180° and ∠B + ∠D = 180°

This calculator applies that rule: enter any one angle, get its opposite. This tutorial covers the theorem, the proof (using the Inscribed Angle Theorem), and how it relates to the converse — a quadrilateral with supplementary opposite angles must be cyclic.

The Inscribed Quadrilateral Theorem

For any quadrilateral ABCD inscribed in a circle:

• ∠A + ∠C = 180° (one pair of opposite angles)
• ∠B + ∠D = 180° (the other pair)

Either equation forces the other, since all 4 angles must sum to 360° regardless.

Why the theorem is true

The proof uses the Inscribed Angle Theorem: an inscribed angle in a circle is half the central angle subtending the same arc.

For cyclic quadrilateral ABCD:

• Angle A is an inscribed angle that intercepts arc BCD (going one way around the circle).
• Angle C is an inscribed angle that intercepts arc DAB (going the other way).
• These two arcs together make up the entire circle = 360°.

Each inscribed angle is half its intercepted arc. So ∠A + ∠C = (arc BCD)/2 + (arc DAB)/2 = (arc BCD + arc DAB)/2 = 360°/2 = 180°. ✓

Same logic for ∠B + ∠D.

The converse

The theorem also works in reverse: if opposite angles of a quadrilateral sum to 180°, the quadrilateral IS cyclic (inscribed in a circle).

So the theorem is an "if-and-only-if":

• Cyclic quadrilateral → opposite angles supplementary
• Opposite angles supplementary → quadrilateral is cyclic

This is a common proof tool: show that some quadrilateral has supplementary opposite angles to conclude it lies on a circle.

Worked example 1 — find opposite angle

Cyclic quadrilateral with ∠A = 110°. Find ∠C.

∠C = 180° − 110° = 70°.

Worked example 2 — all four angles

Cyclic quadrilateral with ∠A = 95°, ∠B = 80°. Find ∠C and ∠D.

∠C = 180° − ∠A = 85°.
∠D = 180° − ∠B = 100°.

Check: ∠A + ∠B + ∠C + ∠D = 95 + 80 + 85 + 100 = 360°. ✓

Worked example 3 — verifying a quadrilateral is cyclic

Given a quadrilateral with angles 90°, 95°, 90°, 85°. Is it cyclic?

Check opposite pair 1: 90° + 90° = 180°. ✓
Check opposite pair 2: 95° + 85° = 180°. ✓

Both pairs are supplementary. By the converse, the quadrilateral IS cyclic.

Quadrilaterals that are always cyclic

• Rectangle: all angles are 90°, so opposite angles sum to 180°. Always cyclic.
• Square: special rectangle. Always cyclic.
• Isosceles trapezoid: the symmetry forces supplementary opposite angles. Always cyclic.
• Right kite: a kite with two opposite right angles. Cyclic.

Quadrilaterals that are NEVER (necessarily) cyclic

• General parallelogram (non-rectangle): opposite angles are EQUAL (not supplementary), so they sum to 2·angle, which is 180° only when angle = 90°. So non-rectangular parallelograms are NOT cyclic.
• Non-square rhombus: a parallelogram, so subject to the same rule. NOT cyclic.
• General kite: may or may not be cyclic depending on its angles.

The diagonal-angle relationship

For a cyclic quadrilateral, the angle between a diagonal and a side equals the angle between the diagonal and the side it subtends across the quadrilateral. (Inscribed Angle Theorem on the same chord.)

This creates many additional angle equalities in cyclic-quadrilateral problems.

Real-world applications

• Olympiad geometry. Cyclic-quadrilateral identities appear in dozens of competition problems.
• Astronomy (historical). Brahmagupta's and Ptolemy's identities for cyclic quadrilaterals supported celestial-sphere calculations.
• Surveying. If four corners of a plot lie on a known circle, the angle theorem helps verify measurements.
• Architecture. Inscribed quadrilateral shapes appear in stained-glass and rose-window designs.

Common mistakes

• Treating opposite angles as equal. In cyclic quadrilaterals, opposite angles are SUPPLEMENTARY (180°), not equal. Opposite angles are equal in PARALLELOGRAMS, which are NOT generally cyclic.
• Forgetting the converse. "Cyclic → supplementary" and "supplementary → cyclic" are both true and used in different problems.
• Applying the rule to non-cyclic quadrilaterals. Most quadrilaterals are not cyclic. Check the supplementary condition first.
• Confusing inscribed quadrilateral with circumscribed. Inscribed = vertices on the circle. Circumscribed = sides tangent to the circle. Different concepts, different theorems.