Compute the area, diagonals, perimeter, angles, height, and side lengths of any parallelogram. Choose between side‑angle input or vertex coordinates. Visualize the parallelogram, its diagonals, height, and vector components on an interactive canvas.
A parallelogram is a quadrilateral with two pairs of parallel opposite sides. This seemingly simple definition gives rise to a rich set of properties: opposite sides are equal in length, opposite angles are equal, and the diagonals bisect each other. The parallelogram is a fundamental shape in Euclidean geometry, appearing everywhere from tiling patterns and structural engineering to vector algebra and computer graphics.
For a parallelogram with side vectors u and v:
Area = |u × v| = |u|·|v|·sin θ
Diagonal 1 = |u + v| , Diagonal 2 = |u − v|
Perimeter = 2(|u| + |v|)
Area: The area of a parallelogram is the magnitude of the cross product of its adjacent side vectors. If base vector u = B − A and side vector v = D − A, then Area = |ux·vy − uy·vx|. Equivalently, Area = base × height, where height = |v|·sin θ.
Diagonals: The diagonals are given by the sum and difference of the side vectors: d₁ = |u + v| (from A to C) and d₂ = |u − v| (from B to D). These lengths satisfy the parallelogram law: d₁² + d₂² = 2(|u|² + |v|²).
Angles: Opposite angles are equal. If θ is the angle at A, then angle at B is 180° − θ. In a parallelogram, adjacent angles are supplementary.
Height: The perpendicular distance between the base and its opposite side is h = b·sin θ, where b is the side length and θ is the interior angle at the base.
| Type | Base (a) | Side (b) | Angle θ | Area | Diagonals | Notes |
|---|---|---|---|---|---|---|
| Rectangle | 5 | 3 | 90° | 15 | √34 ≈ 5.83 | Diagonals equal; θ = 90° |
| Rhombus | 4 | 4 | 60° | 13.86 | √(32±16) ≈ 6.93, 4.00 | All sides equal; diagonals perpendicular |
| Square | 4 | 4 | 90° | 16 | 4√2 ≈ 5.66 | Rectangle + rhombus |
| Obtuse | 5 | 3 | 120° | 12.99 | √(25+9±30·cos120°) ≈ 5.57, 7.00 | One diagonal longer |
Structural engineers often use parallelogram‑based trusses to distribute loads evenly. Consider a steel truss panel with base a = 8 m, side b = 5 m, and angle θ = 75°. The area of the panel is 8 × 5 × sin 75° ≈ 38.64 m², and the diagonals are √(64+25+80·cos75°) ≈ 11.18 m and √(64+25−80·cos75°) ≈ 6.71 m. These diagonal lengths are critical for calculating the required brace lengths and load paths. Our calculator provides these values instantly, allowing rapid iteration of design parameters during the drafting phase.
One of the most important applications of parallelograms is in vector addition. The parallelogram law states that if two vectors u and v are represented as adjacent sides of a parallelogram, then their sum u + v is the diagonal from the common vertex. This principle underpins everything from Newtonian mechanics (resultant forces) to signal processing (phasor addition). Our calculator visualizes these vector relationships directly on the canvas, showing both side vectors and the diagonal sum.
The diagonal lengths also satisfy the identity: d₁² + d₂² = 2(a² + b²), which is a direct consequence of the parallelogram law and provides a quick check for the consistency of any parallelogram.