The quadrilateral calculator above solves a four-sided figure completely: area, perimeter, both diagonals, all four interior angles, and what kind of quadrilateral it turns out to be. It accepts three different sets of starting information, because a quadrilateral is not determined by any one of them alone. Give it four vertex coordinates and everything follows immediately. Give it four sides and one diagonal, or four sides and one angle, and it reconstructs the figure before answering. Give it four sides only, and no honest calculator can answer at all.
Arb Digital built this page because that last point is where most quadrilateral tools quietly cheat. They accept four side lengths, silently assume the shape is cyclic or a parallelogram, and hand back a number that looks authoritative and is simply one of infinitely many possible answers. Saying what extra input is needed is more useful than producing a confident figure for a shape the user never specified, and it takes about one extra field to fix.
Why Four Sides Are Not Enough
A triangle is rigid. Fix three side lengths and the triangle is determined up to reflection — this is the side-side-side congruence rule, and it is why triangulated structures hold their shape. A quadrilateral is not rigid. Pin four rods together at their corners and the frame hinges freely: the sides never change, but the angles, the diagonals and the enclosed area all vary continuously as you push it. A square with sides of 10 can be flattened into a very thin rhombus with sides of 10 and an area approaching zero, without any side changing length at all.
So "four sides of 8, 5.39, 5.10 and 4.12" describes a family of shapes, not a shape. To pick one member of that family you need one more independent measurement. A diagonal works, because it fixes the relative position of two opposite corners and splits the figure into two rigid triangles. One interior angle works, for the same reason from the other direction. A statement that the shape is cyclic — that all four corners lie on a circle — also works, and pins down the unique member of the family with the largest possible area.
That largest-area result is Brahmagupta's theorem and it is worth knowing, because it puts a hard ceiling on any four-sided figure. Among all quadrilaterals with a given set of four sides, the cyclic one encloses the most area. Any tool that returns "the" area from four sides alone is, at best, returning that maximum, and it should say so. Our polygon calculator handles regular polygons and general outlines without the quadrilateral-specific rules this page applies.
How to Use It
- Pick the information you actually have. Coordinates are the most reliable input because they leave nothing ambiguous.
- Enter the corners in order. Going round the outline A to B to C to D matters. Swapping two corners turns the figure into a crossed one with a different area.
- For the sides modes, label consistently. Side a runs from A to B, b from B to C, c from C to D and d from D back to A. Diagonal AC joins the first and third corners.
- Check the classification. The tool names the figure — square, rectangle, rhombus, parallelogram, trapezoid, kite or general — which is a quick sanity check that you entered what you meant.
- Read the angle bars. They should total 360 degrees. A concave figure will have one angle above 180, which is legitimate and worth noticing.
The Formulas and How It's Calculated
From coordinates, area comes from the shoelace formula: sum the cross products xiyi+1 − xi+1yi around the outline and halve the absolute value. With the default corners A(0,0), B(8,0), C(6,5) and D(1,4) the terms are 0, 40, 19 and 0, giving 59, so the area is 29.5. The perimeter is the sum of the four edge lengths: 8 + √29 + √26 + √17 = 8 + 5.3852 + 5.0990 + 4.1231 = 22.6073. The diagonals are the distances AC = √61 = 7.8102 and BD = √65 = 8.0623.
Interior angles come from the dot product at each corner. At A the vectors to D and to B are (1,4) and (8,0); their dot product is 8 and the product of their lengths is 32.985, so the angle is arccos(0.24254) = 75.964 degrees. Repeating at B, C and D gives 68.199, 100.491 and 115.346 degrees, which sum to 360 as they must. Where a corner is reflex the tool detects it from the sign of the cross product relative to the figure's orientation and reports the angle above 180 rather than its supplement.
In the sides-plus-diagonal mode the figure is split by AC into triangles ABC and ACD, each solved by Heron's formula, and the areas added. In the sides-plus-angle mode the angle at A fixes the positions of B and D relative to A, and C is found as the intersection of two circles — one centred on B with radius b, one on D with radius c. That intersection has two solutions, and the tool picks the one that produces a simple, non-crossing outline. Wolfram MathWorld's entry on the quadrilateral sets out the classification and the diagonal relationships in full.
Bretschneider's Formula and the Cyclic Special Case
There is a general area formula for a quadrilateral given four sides and two opposite angles. Bretschneider's formula computes the area as the square root of (s−a)(s−b)(s−c)(s−d) − abcd·cos²(½(A+C)), where s is the semiperimeter and A and C are a pair of opposite interior angles. It is the four-sided cousin of Heron's formula, and the cosine term is exactly the correction that Heron does not need because triangles have no spare freedom to correct for.
Look at what happens when A + C = 180 degrees. The half-sum is 90, its cosine is zero, and the correction term vanishes entirely, leaving √((s−a)(s−b)(s−c)(s−d)). That is Brahmagupta's formula, and the condition A + C = 180 is precisely the condition for the quadrilateral to be cyclic. Since the subtracted term can only reduce the area and is zero only in the cyclic case, the cyclic quadrilateral is provably the largest-area member of the family — the theorem drops straight out of the algebra. The MathWorld entry on Bretschneider's formula gives the diagonal-based form as well.
Convex, Concave and Crossed
A simple quadrilateral is one whose sides do not cross. It is convex if every interior angle is under 180 degrees, which means both diagonals lie inside the figure. It is concave if exactly one angle exceeds 180 — a quadrilateral can have at most one reflex angle, since two would force the total past 360 — and in that case one diagonal falls outside the shape entirely. The dart, a kite pushed inside out, is the familiar example.
A crossed quadrilateral is what you get when the corners are listed out of order, so two sides intersect. The shoelace formula still returns a number for these, but it is the difference between the two loops rather than the enclosed area, and the interior angles no longer sum to 360. This is far and away the most common data-entry error in coordinate geometry, which is why the calculator checks for the crossing explicitly and says so rather than reporting a plausible-looking wrong area. Reordering two corners usually fixes it.
Naming the Shape, and Why the Hierarchy Matters
The families overlap, and the calculator names the most specific one that fits. A square is a rectangle, a rhombus, a parallelogram and a trapezoid all at once; a rectangle is a parallelogram; a parallelogram is a trapezoid under the inclusive definition used here, where a trapezoid needs at least one pair of parallel sides. The exclusive definition, which requires exactly one pair, is also in use and is why textbooks disagree about whether a parallelogram counts as a trapezoid. Neither convention is wrong, but mixing them mid-problem is.
Detection works from measured relationships rather than labels. Both pairs of opposite sides parallel gives a parallelogram; add four right angles for a rectangle; four equal sides for a rhombus; both for a square. Two pairs of adjacent equal sides give a kite. Exactly one pair of parallel sides gives a trapezoid. Because these tests run on floating-point coordinates, the tool uses a small tolerance, so a figure that is a hair off square will still be recognised as one. Dedicated pages exist for the common cases: the trapezoid calculator, the parallelogram calculator and the rectangle area calculator each carry the shortcuts that apply once the shape is known.
Arb Digital builds calculators that state their assumptions instead of hiding them — the kind of page people link to and come back to.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Expecting an area from four sides alone — the shape hinges, so the area is not determined. Supply a diagonal, an angle, or the fact that the figure is cyclic.
- Entering the corners out of order — this produces a crossed outline whose shoelace value is not the enclosed area and whose angles do not sum to 360.
- Assuming both diagonals sit inside the shape — in a concave quadrilateral one diagonal lies outside it, which breaks any argument that relies on splitting the figure at that diagonal.
- Using the parallelogram area formula on a general quadrilateral — base times height only works when the opposite sides really are parallel, and nothing forces them to be.
- Mixing the two trapezoid definitions — decide up front whether a parallelogram counts as a trapezoid, and keep to it, or a classification argument will go in circles.
Related Free Tools From Arb Digital
Handle three-sided figures with the triangle area calculator, measure the outline of any shape with the perimeter calculator, scale figures correctly with the similar triangles calculator, or move the same distance reasoning into three dimensions with the 3D distance calculator. The full free online tools hub lists every geometry calculator we publish.
Frequently Asked Questions
No. Unlike a triangle, a quadrilateral with fixed sides can still flex, so its area varies. You need one more piece of information, such as a diagonal, an interior angle, or the fact that the shape is cyclic.
A diagonal is usually the easiest, because it splits the figure into two rigid triangles that Heron's formula solves directly. An interior angle works just as well if it is easier to measure.
The cyclic arrangement, where all four corners lie on a circle. Brahmagupta's formula gives that maximum, and every other arrangement of the same four sides encloses less.
Yes, for any simple quadrilateral whose sides do not cross, in ordinary plane geometry. If your four angles do not total 360, the outline is crossed or a reflex angle has been recorded as its supplement.
A way of getting the area of any polygon from its vertex coordinates by summing cross products around the outline and halving the absolute value. It requires the corners to be listed in order.
Not a simple one. Two angles above 180 degrees would already exceed the 360-degree total on their own, so at most one interior angle can be reflex.
Yes. The categories are nested, and a square satisfies the defining conditions of both. This calculator reports the most specific name that fits the figure you entered.
This page explains a geometric calculation for educational purposes only, and measured figures should be checked against your own drawing before being used in construction or assessed work.
