Parallelogramm-Satz-Rechner

The Parallelogram Theorem Calculator applies and verifies the complete set of parallelogram theorems — the angle relationships, side relationships, area formulas, and diagonal properties that all follow from "both pairs of opposite sides are parallel". This tutorial walks through each theorem with the proof outline, worked examples, and how the theorems link together as a single network of implications.

The complete parallelogram property catalog

A parallelogram has the following properties, all derivable from the definition (both pairs of opposite sides parallel):

All eight are equivalent in the sense that any quadrilateral with even ONE of properties 1-5 is a parallelogram, and therefore satisfies ALL eight.

The area formula

A = a × b × sin(A)

Where a and b are two adjacent sides and A is the angle between them.

Derivation: drop a perpendicular from one vertex to the opposite side. The height is h = a × sin(A). Then A = base × height = b × h = b × a × sin(A) = ab × sin(A).

Special cases:

• If A = 90° (rectangle): A = a × b × sin(90°) = ab × 1 = ab. Matches the rectangle area formula.
• If a = b (rhombus): A = a² × sin(A).
• If a = b AND A = 90° (square): A = a².

The parallelogram law

For diagonals p and q of any parallelogram with sides a and b:

p² + q² = 2(a² + b²)

The sum of squares of the diagonals equals twice the sum of squares of the sides. This is one of the most elegant identities in plane geometry — it generalizes the Pythagorean theorem.

Verification: in a rectangle, p = q = √(a² + b²). Plugging in: p² + q² = 2(a² + b²). ✓

For a non-rectangle, p ≠ q. The two diagonals split into the formula in different proportions, but their sum of squares is still 2(a² + b²).

Diagonal lengths in terms of sides + angle

By the Law of Cosines applied to the triangle formed by sides a, b and a diagonal:

One diagonal: p² = a² + b² − 2ab·cos(A)
Other diagonal: q² = a² + b² + 2ab·cos(A) (because the other diagonal subtends the supplementary angle 180° − A)

Adding: p² + q² = 2(a² + b²). The cosine terms cancel — recovering the parallelogram law.

Worked example 1 — area from sides and angle

Parallelogram with a = 5, b = 8, A = 60°.

Area = 5 × 8 × sin(60°) = 40 × (√3/2) = 20√3 ≈ 34.64.

Perimeter = 2(5 + 8) = 26.

Diagonal 1: p² = 25 + 64 − 80 × cos(60°) = 89 − 40 = 49 → p = 7.
Diagonal 2: q² = 89 + 40 = 129 → q ≈ 11.36.

Verify parallelogram law: p² + q² = 49 + 129 = 178 = 2(25 + 64). ✓

Worked example 2 — finding a missing diagonal

Parallelogram with sides 6 and 9, and one diagonal of length 10. Find the other diagonal.

By parallelogram law: 10² + q² = 2(36 + 81) = 234.
q² = 234 − 100 = 134
q = √134 ≈ 11.58.

The proof network

The eight parallelogram properties are not independent. Here's how they connect:

From definition (1) → (3) by parallel-line angle theorem: opposite sides being parallel creates alternate-interior or co-interior angle configurations that force opposite angles to match.

From (3) → (2) by congruent triangles: drawing a diagonal creates two triangles. Using the equal opposite angles plus the shared diagonal (reflexive) plus the other angle relations gives congruence by ASA → opposite sides equal.

From (1) and (2) → (5): equal opposite sides + parallel ones combined with vertical angles at the diagonal intersection give congruent sub-triangles by ASA → corresponding parts (the diagonal halves) are equal → bisection.

From (3) → (4): consecutive angles share a side; one is supplementary to the alternate-interior angle of the other (parallel-line theorem). So consecutive angles sum to 180°.

Each property pulls along all the others.

Why the parallelogram law extends Pythagoras

In a rectangle, the two diagonals are equal: p = q. The parallelogram law becomes 2p² = 2(a² + b²), simplifying to p² = a² + b² — the Pythagorean theorem.

So the parallelogram law generalizes Pythagoras. The "correction" comes in when the angle isn't 90° and the two diagonals differ.

Real-world applications

• Vector addition (the parallelogram rule). Adding two vectors graphically creates a parallelogram; their sum is the diagonal. The parallelogram law gives the magnitude of vector sums.
• Force resolution in physics. Forces applied at angles can be combined using the parallelogram of forces.
• Materials engineering. Stress and strain in non-orthogonal axes (e.g., crystals) use parallelogram identities.
• Crystallography. Monoclinic and triclinic crystal lattices have parallelogram-shaped unit cells.

Common mistakes

• Forgetting the sin in the area formula. Area = ab × sin(A), NOT just a × b. Forgetting sin gives the rectangle area instead of the slanted parallelogram area.
• Assuming the diagonals are equal. Only true in rectangles (and squares). General parallelograms have unequal diagonals.
• Confusing parallelogram law with Pythagorean. Parallelogram law involves both diagonals (p² + q²); Pythagorean only one. The two coincide only when p = q.
• Using degree mode when angles are in radians (or vice versa). The sin function gives different results depending on mode. Check calculator setting.