Quadrilateral Area Calculator | Parallelogram, Trapezoid, Rhombus & Kite
Compute the area and perimeter of any quadrilateral: parallelogram, trapezoid, rhombus, kite, rectangle, square, and general quadrilateral by vertex coordinates or side lengths and angles. Includes the Shoelace formula and Brahmagupta's formula for cyclic quadrilaterals.
What Is the Quadrilateral Area Calculator | Parallelogram, Trapezoid, Rhombus & Kite?
A quadrilateral is any polygon with four sides. Different quadrilateral types have specialized area formulas: rectangles and parallelograms use base-height products, trapezoids average the two parallel sides times height, rhombuses and kites use the product of diagonals halved. For any general quadrilateral with known vertex coordinates, the Shoelace (Gauss) formula computes the exact signed area. Brahmagupta's formula gives the maximum area of a cyclic quadrilateral (one inscribed in a circle) given only its four side lengths.
Formula
Rectangle: A = l×w · Parallelogram: A = b×h or a·b·sin(θ) · Trapezoid: A = (a+b)/2 × h · Rhombus/Kite: A = d₁×d₂/2 · Brahmagupta: A = √((s−a)(s−b)(s−c)(s−d)) · Shoelace: A = |Σ(xᵢy_{i+1}−x_{i+1}yᵢ)|/2How to Use
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Select the shape type tab: Rectangle/Square, Parallelogram, Trapezoid, Rhombus, Kite, General (Shoelace), or Cyclic (Brahmagupta).
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Choose the measurement unit from the dropdown: m, cm, ft, in, or yd.
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For Rectangle: enter length and width. Equal values produce a square and the calculator labels it as such.
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For Parallelogram: choose Base × Height mode or Sides & Angle mode (using included angle formula A = a·b·sin θ).
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For Trapezoid: enter both parallel sides (a and b) and the perpendicular height. Add leg lengths c and d for perimeter.
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For General (Shoelace): enter x and y coordinates for all 4 vertices in clockwise or counter-clockwise order.
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For Cyclic (Brahmagupta): enter all four side lengths — the formula gives the maximum possible area for those sides.
Select the quadrilateral type from the shape tabs at the top, choose a measurement unit, then enter the required dimensions. Click Calculate Area to see area and perimeter. For the Shoelace formula, enter all four vertex coordinates in order (clockwise or counter-clockwise).
Example Calculation
Trapezoid: parallel sides a = 12 m, b = 8 m, height h = 5 m, legs c = d = 5.39 m. A = (12+8)/2 × 5 = 10 × 5 = 50 m². P = 12+8+5.39+5.39 = 30.78 m. Cyclic quadrilateral with sides 10, 8, 7, 6: s = (10+8+7+6)/2 = 15.5. A = √((15.5−10)(15.5−8)(15.5−7)(15.5−6)) = √(5.5×7.5×8.5×9.5) = √(3329) ≈ 57.7 m².
Understanding Quadrilateral Area | Parallelogram, Trapezoid, Rhombus & Kite
Area and Perimeter Formulas for All Quadrilateral Types
| Shape | Area Formula | Perimeter Formula | Special Property |
|---|---|---|---|
| Rectangle | A = l × w | P = 2(l + w) | All angles 90°; diagonals equal |
| Square | A = a² | P = 4a | Special rectangle: l = w |
| Parallelogram | A = b × h = a·b·sin(θ) | P = 2(a + b) | Opposite sides parallel and equal |
| Trapezoid | A = (a + b)/2 × h | P = a + b + c + d | Exactly one pair of parallel sides |
| Rhombus | A = d₁ × d₂ / 2 | P = 4a | All sides equal; diagonals perpendicular |
| Kite | A = d₁ × d₂ / 2 | P = 2(a + b) | Two pairs of equal consecutive sides |
| General quad | Shoelace: |Σ(xᵢy_{j} − x_{j}yᵢ)|/2 | Sum of side lengths | Works for any simple polygon |
| Cyclic quad | Brahmagupta: √((s−a)(s−b)(s−c)(s−d)) | P = a + b + c + d | Inscribed in circle; max area |
Hierarchy and Relationships of Quadrilaterals
| Shape | Is a special case of | Additional constraint |
|---|---|---|
| Parallelogram | General quadrilateral | Opposite sides parallel |
| Rectangle | Parallelogram | All angles = 90° |
| Rhombus | Parallelogram | All sides equal |
| Square | Rectangle and Rhombus | All sides equal AND all angles = 90° |
| Trapezoid | General quadrilateral | At least one pair of parallel sides |
| Isosceles trapezoid | Trapezoid | Non-parallel sides equal; cyclic |
| Kite | General quadrilateral | Two pairs of equal adjacent sides |
| Cyclic quad | General quadrilateral | All vertices on a common circle |
Practical Applications
- ›Land surveying: Trapezoids and general quadrilaterals commonly arise in property surveys. The Shoelace formula computes land area from GPS coordinates of corner markers.
- ›Architecture: Trapezoidal cross-sections are used in retaining walls and embankments. Parallelogram layouts appear in slanted roof designs and tiling patterns.
- ›Material cutting: Fabric, sheet metal, and flooring installers need area calculations for irregular quadrilateral pieces to minimize waste and estimate material cost.
- ›Cyclic quadrilaterals in geometry: Cyclic quadrilaterals have Ptolemy's theorem: product of diagonals = sum of products of opposite sides. Useful in geometric proofs and competition mathematics.
- ›Computer graphics: Quadrilateral meshes (quads) are the standard face type in 3D modeling. The Shoelace formula and vertex coordinates underlie polygon rasterization algorithms.
Frequently Asked Questions
What is the Shoelace formula?
The Shoelace (or Gauss area) formula computes the area of any simple polygon from vertex coordinates: A = |Σᵢ(xᵢ·y_{i+1} − x_{i+1}·yᵢ)| / 2, where the sum wraps around (vertex 4 connects back to vertex 1). It works for any simple polygon (non-self-intersecting) regardless of shape. The formula gives signed area — positive for counter-clockwise vertex order, negative for clockwise — hence the absolute value.
What is Brahmagupta's formula?
Brahmagupta's formula, A = √((s−a)(s−b)(s−c)(s−d)) where s = (a+b+c+d)/2, gives the area of a cyclic quadrilateral — one inscribed in a circle — from its four side lengths alone. It generalizes Heron's formula for triangles. For a cyclic quadrilateral, this formula gives the maximum possible area for those four sides. If the quadrilateral is not cyclic, the actual area will be less.
How is a kite different from a rhombus?
A kite has two pairs of equal consecutive sides (adjacent sides equal: top pair equal and bottom pair equal). A rhombus has all four sides equal. Both have perpendicular diagonals. In a kite, only one diagonal bisects the other (the axis of symmetry bisects the cross diagonal). In a rhombus, each diagonal bisects the other. Both have area = d₁ × d₂ / 2.
How do I find the area of a parallelogram using the angle?
If you know sides a and b and included angle θ (the angle between them): A = a · b · sin(θ). This is because the height h = b · sin(θ), and A = base × height = a × b·sin(θ). For θ = 90°, sin(90°) = 1 and the formula reduces to A = a · b, the rectangle case.
What makes a quadrilateral convex vs concave?
A quadrilateral is convex if all interior angles are less than 180° and no diagonal lies outside the figure. Algorithmically, it is convex if the cross products of consecutive edge vectors all have the same sign. A concave quadrilateral has at least one interior angle greater than 180° (a reflex angle), causing one vertex to be "indented." The Shoelace formula still works for concave simple quadrilaterals.
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