Cyclic Quadrilaterals: Concepts, Formulas and Examples

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on a single circle. The circle passing through them is called the circumscribed circle or circumcircle of the quadrilateral. Rectangles, squares, and isosceles trapeziums are always cyclic; a general parallelogram is cyclic only when it is a rectangle.

Cyclic quadrilaterals matter because they convert circle theorems into angle facts: one simple supplementary-angle rule replaces pages of angle chasing. They are a favorite of exam setters precisely because the rule is short but its uses are deep.

Key Concepts

• Opposite angles are supplementary: in a cyclic quadrilateral ABCD, angle A + angle C = 180° and angle B + angle D = 180°. This is the defining property.
• Exterior angle property: the exterior angle at any vertex equals the interior opposite angle. This is a direct consequence of the supplementary rule.
• Converse: if a pair of opposite angles of a quadrilateral sum to 180°, the quadrilateral is cyclic — a circle can be drawn through all four vertices.
• Equal angles in the same segment: angles subtended by the same chord at the circumference are equal, so in cyclic ABCD, angle ABD equals angle ACD (both subtend arc AD).
• Units: angles in degrees; the supplementary sums are always exactly 180°.

Formulas

• Opposite angle rule: ∠A + ∠C = 180° and ∠B + ∠D = 180° for cyclic quadrilateral ABCD in order. This is the only formula you need for most problems.
• Ptolemy's theorem (extension): for a cyclic quadrilateral with sides a, b, c, d in order and diagonals p, q: ac + bd = pq. Useful in advanced problems linking sides and diagonals.
• Circumradius relation: each side satisfies a / sin A = 2R, where R is the circumradius — the same R works for all four sides because one circle passes through all vertices.

Worked Example

ABCD is a cyclic quadrilateral. Angle A = 100° and angle B = 95°. Find angles C and D.

1. Opposite angles are supplementary: angle A + angle C = 180°, so angle C = 180° − 100° = 80°.
2. Similarly, angle B + angle D = 180°, so angle D = 180° − 95° = 85°.
3. Check the total: 100 + 95 + 80 + 85 = 360°, which matches the quadrilateral angle sum. Correct.

Common Misconceptions

• Not every quadrilateral is cyclic. A quadrilateral is cyclic only if its opposite angles are supplementary. A general kite or a non-rectangle parallelogram fails this test.
• Adjacent angles need not sum to 180°. The rule pairs opposite angles (A with C, B with D); students sometimes wrongly add adjacent angles.
• Vertices must be in order around the circle. Labelling a crossing (bow-tie) quadrilateral ABCD breaks the opposite-angle pairing, so always list vertices consecutively around the shape.

Real-World Applications

• Mechanical design: four-bar linkages and rotary mechanisms use cyclic quadrilateral geometry to convert rotation into precise motion.
• Architecture: circular windows divided into four panels (quatrefoils) rely on cyclic quadrilateral proportions for symmetry.
• Sundials and instruments: graduated circular dials use the equal-angles-in-the-same-segment property to mark even divisions.
• Surveying: when four survey stations lie on a circle, the supplementary-angle rule provides a consistency check on measured angles.

A cyclic quadrilateral has all four vertices on one circle, which forces each pair of opposite angles to sum to 180°. That single rule — plus its converse for proving cyclicity — handles nearly every problem on this topic. Just confirm the vertices are in order and pair the angles oppositely, and the answers follow immediately.