Cyclic Quadrilateral Area
Calculator
Results
- Area
- 40.987803
- Semi-perimeter
- 13
- Perimeter
- 26
Results
| Area | 40.987803 |
| Semi-perimeter | 13 |
| Perimeter | 26 |
formula-map diagram
- Area
- 40.987803
- Semi-perimeter
- 13
- Perimeter
- 26
Formula map
Formula
A = √((s−a)(s−b)(s−c)(s−d)), s = (a+b+c+d)/2= 40.987803063838
Note
Simplified model: these are the exact results of the stated idealised geometric formulas, assuming perfect shapes, exact inputs and consistent units. Real measurements carry error, and results such as the cyclic-quadrilateral area only hold when the figure truly satisfies the stated condition. Verify independently before relying on these figures.
More in Advanced geometry
See all →Frequently asked questions
What is a cyclic quadrilateral, and what makes Brahmagupta's formula work for it?+
A cyclic quadrilateral is a four-sided shape where all four vertices lie on a single circle. This special property is exactly what allows Brahmagupta's formula to compute area from the four side lengths alone, without needing any angle measurements.
What is Brahmagupta's formula, and what does 's' represent in it?+
The formula is Area = √((s−a)(s−b)(s−c)(s−d)), where a, b, c, d are the four side lengths and s is the semiperimeter, (a+b+c+d)/2. It's a direct generalization of Heron's triangle-area formula extended to four-sided cyclic shapes.
Why can't I use Brahmagupta's formula for any random quadrilateral with the same four side lengths?+
A set of four side lengths can actually form many different quadrilaterals (think of a rectangle you can 'push' into a slanted parallelogram while keeping the same side lengths), each with a different area. Brahmagupta's formula specifically gives the area of the unique cyclic configuration, which happens to be the maximum possible area for that set of side lengths.
Does Brahmagupta's formula work for a rectangle or square?+
Yes, since rectangles and squares are always cyclic (all four corners touch a circle whenever all angles are 90°), Brahmagupta's formula applies and correctly gives length × width for a rectangle. It also correctly reduces to Heron's triangle formula if one side length is set to zero, treating the quadrilateral as a degenerate triangle.
What happens if the four side lengths I enter can't actually form a valid cyclic quadrilateral?+
If the sides can't close into any valid quadrilateral (for instance, one side is longer than the sum of the other three), one of the (s−x) terms under the square root becomes negative, making the result undefined (an imaginary number). This is the calculator's built-in check that the entered side lengths are geometrically valid.