Cyclic Quadrilateral — Definition, Formula & Examples

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on the circumference of a single circle. Its defining property is that each pair of opposite angles adds up to 180°.

A quadrilateral $ABCD$ is cyclic if and only if there exists a circle passing through all four vertices $A$ , $B$ , $C$ , and $D$ . Equivalently, $ABCD$ is cyclic if and only if its opposite angles are supplementary: $\angle A + \angle C = 180°$ and $\angle B + \angle D = 180°$ .

Key Formula

\angle A + \angle C = 180°\quad\text{and}\quad \angle B + \angle D = 180°

Where:

$\angle A, \angle B, \angle C, \angle D$ = The four interior angles of the cyclic quadrilateral, where A & C are opposite and B & D are opposite

How It Works

To determine whether a quadrilateral is cyclic, check if each pair of opposite angles sums to

180°

. If you know three angles, you can find the fourth using this rule. Another useful test: if an exterior angle of the quadrilateral equals the interior angle at the opposite vertex, the quadrilateral is cyclic. The circle passing through all four vertices is called the circumscribed circle (or circumcircle) of the quadrilateral.

Worked Example

Problem: In cyclic quadrilateral ABCD, angle A = 70° and angle B = 110°. Find angles C and D.

Find angle C: Opposite angles in a cyclic quadrilateral sum to 180°. Angles A and C are opposite.

\angle C = 180° - \angle A = 180° - 70° = 110°

Find angle D: Angles B and D are also opposite, so they must sum to 180° as well.

\angle D = 180° - \angle B = 180° - 110° = 70°

Answer: Angle C = 110° and angle D = 70°.

Why It Matters

Cyclic quadrilaterals appear throughout GCSE and SAT circle theorem questions. Ptolemy's theorem, which relates the sides and diagonals of a cyclic quadrilateral, is a powerful tool in competition geometry and trigonometric proofs.

Common Mistakes

Mistake: Assuming adjacent angles (not opposite angles) must sum to 180°.

Correction: The supplementary relationship applies only to opposite angle pairs. Adjacent angles in a cyclic quadrilateral do not generally sum to 180°.