Quadrilateral Area Calculator (Bretschneider's formula or diagonals)

Understanding the quadrilateral area calculator — concept and principle

This calculator finds the area of any convex quadrilateral, either from its four sides and two opposite angles (Bretschneider's formula) or from its two diagonals and the angle between them. It also shows the perimeter, the semiperimeter and the largest area the four sides could enclose. Use it for land plots, rooms, roofs, garden beds and geometry problems where the shape is not a rectangle.

Sides alone let the frame flex; the 70° and 80° angles fix the shape and its area 39.59.

How it is calculated

Bretschneider: K = √((s − a)(s − b)(s − c)(s − d) − abcd·cos²((α + γ)/2)), with s = (a + b + c + d)/2 and α, γ two opposite angles.
Largest area for these sides (cyclic quadrilateral, α + γ = 180°): Brahmagupta's formula K = √((s − a)(s − b)(s − c)(s − d)).
Diagonals: K = ½·p·q·sin θ.

Sides are entered in order around the shape. Only the inputs of the chosen method are used.

The principle

Four side lengths alone do not fix a quadrilateral: like a hinged frame it can flex, and its area changes as it does. One more piece of information, the sum of two opposite angles, fixes the shape, and Bretschneider's formula combines it with the sides. The cosine term is zero when the opposite angles add to 180°, which is when the vertices lie on a circle; that is the shape with the largest area, giving Brahmagupta's formula. The diagonal formula comes from splitting the quadrilateral into four triangles around the crossing point.

Worked example

Default inputs: a = 5, b = 6, c = 7, d = 8, α = 70°, γ = 80°.

s = (5 + 6 + 7 + 8) ÷ 2 = 13.
(s − a)(s − b)(s − c)(s − d) = 8 × 7 × 6 × 5 = 1,680.
abcd = 1,680; (α + γ)/2 = 75°, cos² 75° ≈ 0.066987, so the correction is ≈ 112.54.
K = √(1,680 − 112.54) ≈ 39.5912.
Perimeter 26; largest possible area √1,680 ≈ 40.9878.

Each point is the calculator re-run with only “Side b” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Side b	Area
1.5	22.8114
3	28.5293
4.5	34.1947
6	39.5912
7.5	44.5184
9	48.7686
12	54.2487

How to read the result

The area is in square units of your side lengths: metres give square metres. The closer your area is to the cyclic maximum, the closer the shape is to having its corners on a circle. If you only know the four sides, you cannot get a single answer; you need an angle or a diagonal.

Common mistakes

Entering sides out of order around the shape.
Using two adjacent angles instead of two opposite angles.
Using the four sides alone and assuming the Brahmagupta value is the area.
Mixing degrees with radians.
Applying the formula to a concave (dented) quadrilateral.

Keep in mind: For land and building work, use a proper survey. Small errors in angles or sides can change the area noticeably, and legal areas must come from qualified surveyors.

Frequently asked questions

Can I find the area of a quadrilateral from four sides only?

No. The shape can flex, so you also need an angle or a diagonal.

What is a cyclic quadrilateral?

One whose four vertices lie on a circle; its opposite angles add up to 180°.

Does Bretschneider's formula work for rectangles?

Yes. All angles are 90°, the cosine term is 0, and the result is length × width.

Sources: Weisstein, E. W., "Bretschneider's Formula", Wolfram MathWorld

Checked 9 Nov 2026. Weisstein, E. W., “Bretschneider's Formula”, Wolfram MathWorld, eq. (3): K = √((s − a)(s − b)(s − c)(s − d) − abcd·cos²[½(A + B)]) for sides a, b, c, d, semiperimeter s and two opposite angles A, B. The diagonal form K = ½·pq·sin θ is the standard formula.

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