Cyclic Quadrilateral Calculator (Brahmagupta's Formula)

What this tool does

When a quadrilateral can be drawn inside a circle with all four corners on it, its opposite angles add to 180° and four sides are enough to fix everything else. Brahmagupta's formula for the area looks like Heron's for a triangle, with a fourth side: √((s−a)(s−b)(s−c)(s−d)). Of all quadrilaterals with the same four sides, the cyclic one encloses the largest area. Any four lengths work as long as the longest is shorter than the other three added together; the order of the sides around the figure matters for the angles and diagonals, not for the area.

Formula

K = √((s−a)(s−b)(s−c)(s−d)) · p = √((ac+bd)(ad+bc) ÷ (ab+cd)) · q = √((ac+bd)(ab+cd) ÷ (ad+bc)) · pq = ac + bd · R = √((ab+cd)(ac+bd)(ad+bc)) ÷ 4K

Variables

Worked example

Limitations

• Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.
• The calculation runs at full precision and only the display is rounded. If you copy an intermediate value and retype it, small differences can appear.