Cyclic Quadrilateral Calculator

Name: Cyclic Quadrilateral - Geometry Calculator
Author: Jürgen Kummer

Calculations at a cyclic quadrilateral. A cyclic quadrilateral is a quadrangle whose vertices lie on a circle, the sides are chords of the circle.
Enter the four sides (chords) a, b, c and d, choose the number of decimal places and click Calculate. Angles are calculated and displayed in degrees, here you can convert angle units.

Formulas:

e = \sqrt{\frac{(a c + b d) \cdot (a d + b c)}{(a b + c d)}}

f = \sqrt{\frac{(a b + c d) \cdot (a c + b d)}{(a d + b c)}}

a \cdot c + b \cdot d = e \cdot f (Ptolemy's theorem)

α = \arccos (\frac{a^{2} + d^{2} - b^{2} - c^{2}}{2 \cdot (a d + b c)})

δ = \arccos (\frac{d^{2} + c^{2} - a^{2} - b^{2}}{2 \cdot (d c + a b)})

β = 180 ° - δ

γ = 180 ° - α

p = a + b + c + d

Semiperimeter

s = \frac{p}{2}

A = \sqrt{(s - a) \cdot (s - b) \cdot (s - c) \cdot (s - d)}

r_{c} = \frac{1}{4 \cdot A} \cdot \sqrt{(a b + c d) \cdot (a c + b d) \cdot (a d + b c)}

Side lengths, diagonals, perimeter and radius have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

The center of the circle is at the intersection of two perpendicular bisectors. These can be constructed by drawing intersecting circles around two neighboring vertices and connecting their intersections.

Cyclic quadrilateral, perimeter and area

perimeter p, area A

sides and angles

radius and circumcircle

perpendicular bisectors

A chord is the straight line connecting two points on a curve, whereby this curve is often, as here, a circular path. The cyclic quadrilateral is made up of four chords, two of which start at a point on the circle without these chords intersecting. Ptolemy's theorem, which refers to the lengths of the sides and the two diagonals in the chord quadrilateral, can be understood as a generalization of the much more well-known Pythagorean theorem. It is written out as follows: the product of the two diagonal lengths is equal to the sum of the products of the both opposite side lengths. This theorem was formulated by Claudius Ptolemy, who worked in Alexandria in Egypt in the second century and was mainly concerned with astronomy and geometry. His geocentric worldview, in which the earth is at the center and is orbited by the sun and the other planets, was in use for centuries, but was eventually proven to be fundamentally wrong. Ptolemy's theorem, on the other hand, is still correct.

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