Cyclic Quadrilateral Calculator

Work out cyclic quadrilateral instantly with clear inputs, formula shown and shareable results.

About Cyclic Quadrilateral Calculator

Brahmagupta's formula gives the maximum area for four given side lengths, achieved exactly when the quadrilateral is cyclic: A = √[(s−a)(s−b)(s−c)(s−d)]. The circumradius follows from Parameshvara's formula.

Formulas

Brahmagupta

A = √[(s−a)(s−b)(s−c)(s−d)], s = (a+b+c+d)/2

Parameshvara

R = √[(ab+cd)(ac+bd)(ad+bc)] / (4A)

Worked example

With the default values below, Cyclic Quadrilateral Calculator returns Area (Brahmagupta) of 40.9878. Change any input above and every figure here is recalculated in your browser.

Inputs used

Side aminimum 0: 5
Side bminimum 0: 6
Side cminimum 0: 7
Side dminimum 0: 8

Results

Area (Brahmagupta): 40.9878
Semiperimeter s: 13.0000
Circumradius R: 4.6664
Perimeter: 26.0000

Step by step

1Values used
2Brahmagupta
3Parameshvara
4Area (Brahmagupta)= 40.9878
5Semiperimeter s= 13.0000
6Circumradius R= 4.6664
7Perimeter= 26.0000

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What makes a quadrilateral cyclic?

All four vertices lie on one circle, which happens exactly when opposite angles sum to 180°.

Is this the same as Heron's formula?

It is the four-sided generalisation. Setting d = 0 collapses Brahmagupta's formula back to Heron's.