Quadrilateral Calculator - Sides, Angle, And Cyclic Type

A quadrilateral calculator is a single-form tool that turns four side lengths and the sum of two opposite angles into the area, perimeter, semiperimeter, and cyclic type of a general four-sided figure. It applies Bretschneider's formula to the convex case and Brahmagupta's formula when the figure is cyclic, so the same inputs work for rectangles, kites, surveyed plots, and every shape in between.

• • Geometry homework: Confirm the area of a general convex quadrilateral from four side lengths and one angle, without re-deriving Bretschneider's formula by hand.
• • Surveying irregular plots: Convert a field book listing four side lengths and the angle at one corner into an area in square meters or square feet.
• • Checking the cyclic case: Test whether a four-sided figure is cyclic by setting the opposite-angle sum to 180 and reading the maximum area label.
• • Comparing four-sided shapes: Use the same form for rectangles, kites, and irregular quadrilaterals by changing only the side inputs and the angle sum.

The strength of a quadrilateral calculator is that the four sides and one angle are the only inputs; the type label and the active formula are outputs, so the same form covers a square, a kite, and a survey of a lopsided yard.

When the four sides belong to a regular polygon and you want the area from the apothem, the next tool in the layout is the Area Quadrilateral Calculator.

The form reads the four side lengths and the sum of two opposite angles, then computes the semiperimeter, the Brahmagupta term, and the cosine correction, before applying the rule that matches the input. Cyclic quadrilaterals skip the correction because the cosine of half of 180 degrees is zero; general convex figures keep the full Bretschneider correction.

• a, b, c, d: The four side lengths, in the same linear unit.
• s: Semiperimeter, equal to (a + b + c + d) / 2.
• alpha + gamma: Sum of two opposite interior angles, in degrees; the same formula also works with beta + delta.
• cos^2(theta / 2): Cosine correction; this term is zero when the angle sum is 180, so the figure is cyclic.

The unit selector at the top of the form scales every input at once, and the result panel carries the matching square unit so the area reads as square meters, square feet, or any other unit you choose.

According to Wolfram MathWorld, the area of a general convex quadrilateral with sides a, b, c, d and opposite-angle sum theta is A = sqrt((s-a)(s-b)(s-c)(s-d) - a*b*c*d*cos^2(theta/2))

The Polygon Area Calculator covers regular polygons and other multi-sided figures where the side lengths come from a closed shape with a known number of sides.

A small set of terms comes up every time you read or apply the area formulas. Knowing which one applies to your figure decides which formula the calculator uses.

When the four sides belong to a real rectangle, the two pairs of opposite angles each sum to 180, so the figure is cyclic and Brahmagupta's formula reduces to the length-times-width product.

Splitting a quadrilateral along a diagonal turns the same four sides into two triangles, and the Triangle Calculator is the right tool when you only need the area of one of those pieces.

Type the four side lengths in any order, set the sum of two opposite angles, and pick the unit you measured with. The result panel updates as you type, so the area, perimeter, semiperimeter, and type are all visible on the same screen.

1. 1 Enter the four side lengths: Use the same linear unit on every side. Mixed units give an area wrong by the square of the conversion factor.
2. 2 Set the sum of two opposite angles: Use 180 for a cyclic quadrilateral, including rectangles, squares, and any figure inscribed in a circle. Smaller values mean a general convex shape; 0 or 360 collapses the figure to a line.
3. 3 Pick the length unit: The selector at the top of the form scales the area result. The unit label on the area is the matching square form, so feet surface as square feet.
4. 4 Read the area and the cyclic type: The primary result is the area, the second line is the maximum area for a cyclic arrangement, and the type label tells you whether your inputs describe a cyclic or general convex figure.
5. 5 Adjust the inputs if the type reads Degenerate: A zero side or an angle sum that makes the radicand negative triggers the Degenerate label. Change the side or the angle to produce a positive radicand and read the result again.

A surveyor measures a plot of 4 m, 5 m, 6 m, and 7 m on each side, with the two angles at the first vertex summing to 90 degrees. Enter the four sides in any order, set the angle sum to 90, and the calculator reports 20.49 square meters, a max area of 28.98 square meters for a cyclic arrangement, and a type label of General convex quadrilateral.

When the figure is actually a rectangle, square, triangle, or circle rather than a general four-sided shape, the Area Calculator runs the matching per-shape formula in a single form.

The tool replaces two separate area formulas and a separate cyclic test with a single form. Once the four sides and the angle sum are in, the rest of the work happens in one place.

• • Bretschneider and Brahmagupta in one form: The same form covers a rectangle, a kite, and an irregular four-sided figure, with the cosine correction switched on or off based on the angle sum you enter.
• • Cyclic type label on the result: The result panel names whether the figure is cyclic, general convex, or degenerate, so you do not have to remember which formula applies to which shape.
• • Max area as a second reading: The second result line is the Brahmagupta maximum for the four sides, which is the upper bound a cyclic arrangement reaches. Comparing the live area against this bound is a quick check on the input.
• • Perimeter and semiperimeter at the same time: Side lengths and a per-side unit also produce the perimeter and the semiperimeter, which are the building blocks of the area formulas and the next inputs for many other geometry tools.
• • Pairs with related shape work: The trapezoid case, the regular-polygon case, and the per-shape rules use the same area unit and the same input style, so the quadrilateral result carries over to those tools without reformatting.

When the figure in front of you is a trapezoid with one pair of parallel sides, the Area Of A Trapezoid Calculator gives the same perimeter and area with a smaller input set.

Most mistakes with this kind of calculation come from the wrong angle sum or a side measured in a different unit, not from the formula itself. A short checklist before reading the result keeps the area honest.

• • A zero side, a 360-degree angle sum, or any input that drives the Bretschneider radicand below zero collapses the figure to a line. The calculator clamps the area to zero and labels the figure Degenerate.
• • Two different convex quadrilaterals can share the same four side lengths and the same opposite-angle sum, so the formula returns the same area for both. A diagram and an extra side or diagonal will disambiguate the figure before you compute.

According to Math Open Reference, a quadrilateral is a closed polygon with exactly four straight sides and the sum of its interior angles is always 360 degrees