Quadrilateral Calculator

Enter a quadrilateral's four sides, its two diagonals, and the angle between the diagonals to get the perimeter (P = a+b+c+d) and area (A = ½d₁d₂sinθ).

Quick Facts

Perimeter formula
P = a + b + c + d
Simply the sum of the four side lengths.
Area formula (diagonals)
A = ½ × d₁ × d₂ × sin θ
Works for any quadrilateral, convex or concave, using the angle between the diagonals.
Quadrilateral inequality
longest side < sum of other three
A necessary condition for four lengths to close into a simple quadrilateral.
Angle sum
Interior angles = 360°
True for any simple (non-self-intersecting) quadrilateral.

Your Results

Calculated
Area
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A = ½ × d₁ × d₂ × sin θ
Perimeter
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P = a + b + c + d
Semi-perimeter
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s = P / 2
Shape Check
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Quadrilateral inequality test

Ready

Enter the four sides, both diagonals, and the angle between them, then press Calculate.

Formula and Method for the Quadrilateral Calculator

A quadrilateral is any four-sided polygon: a, b, c, and d are its side lengths, and its two diagonals connect opposite corners. Unlike a triangle, four side lengths alone do not fix a quadrilateral's shape — it can flex like a hinge, so the same sides can enclose very different areas. To get a single, unambiguous area, this calculator uses the two diagonals and the angle between them: A = ½ × d₁ × d₂ × sin θ. Perimeter comes directly from the four sides: P = a + b + c + d.

How the calculation works

Enter the four side lengths, the lengths of both diagonals (d₁ and d₂), and the angle θ between them where they cross, then choose the unit. The calculator adds the sides for the perimeter, halves that for the semi-perimeter (s = P/2), and multiplies the diagonals by sin θ and by ½ for the area. This diagonal formula is exact for any simple quadrilateral, convex or concave — it is the same relationship that gives A = ½d₁d₂ for a rhombus or kite, where the diagonals happen to cross at a right angle (sin 90° = 1). Before computing area, the tool also checks the quadrilateral inequality: the longest of the four sides must be shorter than the sum of the other three, or the sides cannot close into a real quadrilateral at all.

Common mistakes

  • Assuming sides alone determine area: a quadrilateral is not rigid like a triangle — you need the diagonals and the angle between them (or another equivalent piece of information) to pin down a unique area from four side lengths.
  • Angle out of range: the angle between diagonals must be strictly between 0° and 180°; at 0° or 180° the diagonals are collinear and the "quadrilateral" collapses to a line with zero area.
  • Mixing units: enter all four sides and both diagonals in the same unit — converting only some of them skews both the perimeter and the area.
  • Ignoring the shape check: if the longest side is not shorter than the sum of the other three, no quadrilateral with those exact side lengths can exist, regardless of what the diagonals say.

Real-world applications

  • Land surveying uses this diagonal method to find the area of irregular four-sided plots without needing right angles.
  • Carpentry and framing use the perimeter for trim or edging material, and the diagonal/angle relationship to verify a panel is square (equal diagonals crossing at 90° confirm a rectangle).
  • CAD, drafting, and roofing layouts use the same formula for kite- and diamond-shaped panels and trusses.
  • Tiling or paving an irregular quadrilateral patio or lot uses the area result to estimate material quantities.

Frequently Asked Questions

What is the formula for the area of a quadrilateral?
For any quadrilateral, the area can be found from its two diagonals and the angle between them: A = ½ × d₁ × d₂ × sin θ. This formula works for convex and concave quadrilaterals alike, and it reduces to the familiar rhombus/kite formula A = ½d₁d₂ when the diagonals are perpendicular (θ = 90°, sin θ = 1).
How do I find the perimeter of a quadrilateral?
Add the four side lengths together: P = a + b + c + d. For example, sides of 5, 6, 7, and 8 ft give a perimeter of 26 ft.
Why do I need diagonals and an angle, not just the four sides, to find the area?
Unlike a triangle, a quadrilateral is not rigid — the same four side lengths can be hinged into many different shapes with different areas, like squashing a square into a thin parallelogram. You need at least one more piece of information, such as the diagonals and the angle between them, to pin down a unique area.
What is the quadrilateral inequality, and why does the calculator check it?
Just like the triangle inequality, four lengths can only form a simple quadrilateral if the longest side is shorter than the sum of the other three. If that check fails, the sides you entered cannot be closed into a valid four-sided shape.

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