Quadrilateral Calculator

A quadrilateral is a polygon with exactly four sides, four vertices, and four interior angles. The sum of interior angles is always 360°. Examples include squares, rectangles, trapezoids, parallelograms, rhombuses, and kites.

For a general quadrilateral with diagonals d₁ and d₂ meeting at angle θ: A = ½ × d₁ × d₂ × sin(θ). For a trapezoid: A = ½(b₁ + b₂) × h. For coordinates, use the shoelace formula.

Given vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), (x₄,y₄), the area is A = ½|x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁ - y₁x₂ - y₂x₃ - y₃x₄ - y₄x₁|. This works for any simple quadrilateral.

The sum of interior angles is always 360° = (n-2) × 180° with n = 4. In a parallelogram, opposite angles are equal. In a cyclic quadrilateral, opposite angles sum to 180°.

The perimeter is P = a + b + c + d, where a, b, c, and d are the four side lengths. For special quadrilaterals: square P = 4s, rectangle P = 2(l+w), rhombus P = 4s.

A cyclic quadrilateral has all four vertices on a circle. Its opposite angles sum to 180°. The area can be computed using Brahmagupta's formula: A = √((s-a)(s-b)(s-c)(s-d)), where s is the semiperimeter.

Brahmagupta's formula gives the area of a cyclic quadrilateral: A = √((s-a)(s-b)(s-c)(s-d)), where s = (a+b+c+d)/2 is the semiperimeter and a, b, c, d are the side lengths.

For a general quadrilateral, use the law of cosines on the triangles formed by the diagonals. For special types: rectangle diagonal = √(l²+w²), parallelogram uses the parallelogram law: d₁² + d₂² = 2(a² + b²).

The main types are: square (4 equal sides, 4 right angles), rectangle (opposite sides equal, 4 right angles), parallelogram (opposite sides parallel and equal), rhombus (4 equal sides), trapezoid (one pair of parallel sides), and kite (two pairs of equal adjacent sides).

A trapezoid has exactly one pair of parallel sides (in the exclusive definition). A parallelogram has two pairs of parallel sides. Every parallelogram is a special case of a trapezoid under the inclusive definition.

Use the shoelace formula: A = ½|Σ(xᵢyᵢ₊₁ - xᵢ₊₁yᵢ)|. List vertices in order (clockwise or counterclockwise), apply the cross-multiplication pattern, and take half the absolute value.

A concave quadrilateral has one interior angle greater than 180° (a reflex angle). The quadrilateral "caves inward." One diagonal lies outside the shape. Area can still be calculated using the shoelace formula.

For fixed side lengths, the maximum area occurs when the quadrilateral is cyclic (inscribed in a circle). The maximum area is given by Brahmagupta's formula.

In a parallelogram, the diagonals bisect each other. The sum of squares of the diagonals equals the sum of squares of all four sides: d₁² + d₂² = 2(a² + b²). In a rectangle, diagonals are equal.

The area of a kite is A = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals. The diagonals of a kite are perpendicular, making this formula straightforward.

A self-intersecting (crossed) quadrilateral has sides that cross each other, forming a bowtie shape. The shoelace formula gives a signed area where the crossing creates positive and negative regions.

Use the angle sum property: all four angles sum to 360°. If three angles are known, the fourth is 360° minus their sum. For cyclic quadrilaterals, opposite angles sum to 180°.

The Varignon parallelogram is formed by connecting the midpoints of any quadrilateral's sides. It is always a parallelogram, regardless of the original quadrilateral's shape. Its area is half the original.

Surveyors use the shoelace formula or triangulation to compute land area from boundary coordinates. Quadrilateral area calculations are fundamental to land measurement and property boundary definitions.

Yes, a rhombus has all equal sides but does not require right angles. A square is a special rhombus with 90° angles. A rhombus has equal sides and opposite angles equal, but adjacent angles may differ.

The area of a parallelogram is A = b × h, where b is the base and h is the perpendicular height. This is the same as a rectangle with the same base and height, even though the sides are slanted.

Classify by checking: Are all sides equal? (Square or rhombus) Are opposite sides parallel? (Parallelogram, rectangle, rhombus, square) Are there right angles? (Rectangle or square) Does it have one pair of parallel sides? (Trapezoid).

The Quadrilateral Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Quadrilateral Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.