Cyclic Quadrilateral Calculator

A cyclic quadrilateral is a quadrilateral whose four vertices all lie on a single circle. This is the quadrilateral analog of "any triangle can be inscribed in a circle" — but unlike triangles, not every quadrilateral is cyclic. The cyclic property creates several remarkable relationships: opposite angles sum to 180°, the area follows Brahmagupta's formula (the cyclic generalization of Heron's), and the diagonals satisfy Ptolemy's theorem. This tutorial covers all three.

The defining property

A quadrilateral ABCD is cyclic if and only if all four vertices lie on a single circle. The circle is called the circumscribed circle (or circumcircle), and its radius is the circumradius R.

Cyclic quadrilateral test — the simplest version: opposite angles sum to 180°.

∠A + ∠C = 180°
∠B + ∠D = 180°

(Either equation implies the other, since all four angles sum to 360° in any quadrilateral.)

Why opposite angles sum to 180°

This is a direct consequence of the Inscribed Angle Theorem: an inscribed angle in a circle is half the central angle that subtends the same arc.

For cyclic quadrilateral ABCD: angles A and C are inscribed angles subtending opposite arcs of the circle. The two arcs together form the entire circle (360° of arc). Each inscribed angle is half its corresponding arc, so their sum is half of 360° = 180°.

The reverse holds too (converse): if opposite angles of a quadrilateral sum to 180°, then it IS cyclic. This is the practical test for "is this quadrilateral cyclic?".

Brahmagupta's formula

The area of a cyclic quadrilateral with side lengths a, b, c, d is:

Area = √((s − a)(s − b)(s − c)(s − d))

where s = (a + b + c + d) / 2 is the semi-perimeter.

Discovered by the Indian mathematician Brahmagupta in the 7th century, this formula is the cyclic quadrilateral analog of Heron's formula for triangles. In fact, if you let one of the sides shrink to length 0, the cyclic quadrilateral degenerates into a triangle and Brahmagupta's formula reduces to Heron's.

For NON-cyclic quadrilaterals, this formula gives an OVER-estimate of the area. Brahmagupta's value is the maximum area achievable for a quadrilateral with the four given side lengths — and that maximum is realized only when the quadrilateral is cyclic.

Worked example — Brahmagupta's formula

Cyclic quadrilateral with sides 3, 4, 5, 6.

s = (3 + 4 + 5 + 6) / 2 = 9.

Area = √((9−3)(9−4)(9−5)(9−6)) = √(6 × 5 × 4 × 3) = √360 ≈ 18.97.

Ptolemy's theorem

For cyclic quadrilateral ABCD with diagonals p = AC and q = BD:

p × q = a × c + b × d

The product of the diagonals equals the sum of the products of opposite sides (where a = AB, c = CD are one pair of opposite sides, and b = BC, d = DA are the other).

Ptolemy's theorem is one of the most elegant in geometry. It gives a direct relationship between diagonals and sides of cyclic quadrilaterals.

Ptolemy's in a rectangle

A rectangle is a cyclic quadrilateral (all four vertices lie on a circle whose diameter is the rectangle's diagonal). For a rectangle with sides l and w:

• a = c = l (the two pairs of opposite sides are equal)
• b = d = w
• Both diagonals are equal: p = q = √(l² + w²)

Ptolemy's: p² = a×c + b×d = l² + w². Which is just the Pythagorean theorem! Ptolemy's theorem generalizes the Pythagorean theorem to all cyclic quadrilaterals.

Worked example — Ptolemy's

Cyclic quadrilateral with sides AB = 5, BC = 6, CD = 8, DA = 7. Diagonal AC = 9. Find diagonal BD.

By Ptolemy's: AC × BD = AB × CD + BC × DA
9 × BD = 5 × 8 + 6 × 7 = 40 + 42 = 82
BD = 82/9 ≈ 9.11.

Which quadrilaterals are always cyclic?

Several special quadrilaterals are guaranteed cyclic:

• Rectangle: all angles are 90°, so opposite angles sum to 180°. Always cyclic.
• Square: a special rectangle. Cyclic.
• Isosceles trapezoid: always cyclic. The equal legs force opposite angles to be supplementary.
• Right kite: a kite with two opposite right angles. Cyclic.

NOT necessarily cyclic:

• General parallelogram: opposite angles are equal, so they sum to 2×angle. Only equals 180° if angle = 90°, which makes it a rectangle. Non-rectangular parallelograms are NOT cyclic.
• Rhombus: a parallelogram with equal sides. Non-square rhombuses are NOT cyclic.
• General trapezoid: may or may not be cyclic depending on whether it's isosceles.
• General kite: may or may not be cyclic.

Real-world applications

• Astronomy (historical). Brahmagupta and Ptolemy both developed cyclic-quadrilateral identities to support astronomical calculations involving celestial positions on the "celestial sphere".
• Architecture. Inscribed quadrilateral shapes appear in stained glass, mosaic, and circular window designs.
• Olympiad mathematics. Cyclic quadrilateral identities (Ptolemy, Brahmagupta, opposite-angle property) appear in dozens of competition problems.
• Computer graphics. Detecting whether four detected points lie on a circle uses the cyclic condition.

Common mistakes

• Applying Brahmagupta's formula to non-cyclic quadrilaterals. The formula only works for cyclic quadrilaterals. For a general quadrilateral with sides a, b, c, d, the area depends on more than just side lengths — diagonal lengths or angles are needed.
• Confusing Ptolemy's theorem direction. Ptolemy gives "product of diagonals = sum of products of opposite sides", NOT "diagonal = sum of opposite sides" or other rearrangements.
• Treating any parallelogram as cyclic. Only rectangles (parallelograms with right angles) are cyclic. General parallelograms and rhombuses are not.
• Forgetting the "if and only if" in the opposite-angles rule. Opposite angles summing to 180° is BOTH necessary (every cyclic quadrilateral has this) AND sufficient (a quadrilateral with this property is cyclic). The biconditional gives you the cyclic-or-not test.