Quadrilateral Shape Calculator

A quadrilateral is any closed plane figure with four straight sides. That single definition covers an enormous family: squares, rectangles, parallelograms, rhombuses, trapezoids, kites, and "fully irregular" four-sided shapes with no special properties. This calculator auto-detects which family your four sides + one angle define, applies the correct area formula, and returns perimeter, diagonals, and angle properties. This tutorial covers the decision tree the calculator uses internally + 3 worked examples spanning the common cases.

Universal property: interior angles sum to 360°

Every quadrilateral — regardless of family — has interior angles summing to exactly 360°. This follows from splitting any quadrilateral into two triangles via a diagonal: each triangle's angles sum to 180°, so two triangles → 360°. The corollary: knowing 3 of the 4 interior angles always gives the 4th by subtraction.

The 6 named families (decision tree)

1. Square: all 4 sides equal AND all 4 angles 90°. Area = s². Diagonals equal and perpendicular.
2. Rectangle: opposite sides equal AND all 4 angles 90°. Area = length × width. Diagonals equal (but not perpendicular unless it's also a square).
3. Rhombus: all 4 sides equal but angles not 90°. Area = ½ × d₁ × d₂ (product of diagonals divided by 2). Diagonals perpendicular bisectors of each other.
4. Parallelogram: opposite sides parallel AND equal. Area = base × height. Diagonals bisect each other but are not equal.
5. Trapezoid: exactly one pair of parallel sides (the bases). Area = ½(b₁ + b₂) × h. See the dedicated Trapezoid Calculator for sub-types.
6. Kite: two pairs of adjacent (not opposite) equal sides. Area = ½ × d₁ × d₂. Diagonals perpendicular; one bisects the other.

If none of the above match, the calculator falls back to irregular quadrilateral and applies Brahmagupta's formula (for cyclic quadrilaterals where opposite angles sum to 180°) or the Shoelace formula (when vertex coordinates are given).

Worked Example 1 — Rectangle

Input: a = 8, b = 5, c = 8, d = 5, A = 90°.
Detection: opposite sides equal (a=c, b=d) + one angle = 90° → rectangle.
Area = a × b = 40. Perimeter = 2(a + b) = 26. Diagonal = √(a² + b²) = √89 ≈ 9.43.

Worked Example 2 — Rhombus

Input: a = 6, b = 6, c = 6, d = 6, A = 60°.
Detection: all sides equal + angle ≠ 90° → rhombus.
One diagonal d₁ = 2 × a × sin(A/2) = 2 × 6 × sin(30°) = 6. Other diagonal d₂ = 2 × a × cos(A/2) = 2 × 6 × cos(30°) ≈ 10.39. Area = ½ × d₁ × d₂ ≈ 31.18 (or via a² × sin(A) = 36 × sin(60°) ≈ 31.18 — both formulas agree).

Worked Example 3 — General Parallelogram

Input: a = 10, b = 6, c = 10, d = 6, A = 70°.
Detection: opposite sides equal but angle ≠ 90° → parallelogram (not rectangle).
Area = a × b × sin(A) = 60 × sin(70°) ≈ 56.38. Perimeter = 2(a + b) = 32. Height h = b × sin(A) ≈ 5.64 (perpendicular distance from one a-side to the opposite a-side).

Common mistakes

• Treating a rhombus as a square. All four sides equal doesn't mean square — a square also requires all angles = 90°. Check the angle field; a rhombus with 60°/120° angles is NOT a square.
• Using "base × height" for a parallelogram with the slanted side. The height in A = b × h must be PERPENDICULAR to the base, not the slanted side a. To get h from a slanted side: h = a × sin(angle to base).
• Confusing kite and rhombus. Kite has two pairs of ADJACENT equal sides (a=b and c=d). Rhombus has all 4 equal. They look superficially similar.
• Assuming opposite angles are equal. Only true in parallelograms (and special cases like rectangle / rhombus / square). Trapezoids and kites generally have all 4 different angles.

When to use a different calculator

• For trapezoids (one parallel pair), the Trapezoid Calculator handles sub-types more accurately.
• For parallelograms specifically (and finding angles from a single known angle), use the Parallelogram Angle Solver.
• For 4 vertex coordinates (irregular quadrilaterals), use the Quadrilateral with Points tool which applies the Shoelace formula.
• For cyclic quadrilaterals (inscribed in a circle), the Inscribed Quadrilateral Calculator uses Brahmagupta's formula and adds the cyclic-specific angle relationships.