Parallelogram Sides Calculator

The Parallelogram Sides Calculator solves the reverse-direction problem of standard parallelogram calculations: given both diagonal lengths and the angle between them, find the two side lengths. This uses the geometric fact that the diagonals of a parallelogram bisect each other, so the intersection point creates 4 triangles whose sides can be computed via the Law of Cosines. This tutorial walks through the derivation, three worked examples, and how this connects to the parallelogram law.

The setup

A parallelogram ABCD has two diagonals AC and BD that intersect at point M. By the parallelogram property:

• M is the midpoint of both diagonals (each is bisected).
• So AM = MC = d₁/2 and BM = MD = d₂/2.

Let the angle between the diagonals at M be θ. Then the 4 sub-triangles formed by the diagonals all have one of their vertex angles at M equal to either θ or 180° − θ.

Computing the sides

Take triangle ABM. Its sides at M are AM = d₁/2 and BM = d₂/2, with angle θ between them (or 180° − θ — depends on which triangle). The opposite side of M is AB, which equals one of the parallelogram's sides.

By the Law of Cosines on triangle ABM:

AB² = (d₁/2)² + (d₂/2)² − 2(d₁/2)(d₂/2)cos(θ)
= d₁²/4 + d₂²/4 − (d₁d₂/2)cos(θ)

So AB = ½√(d₁² + d₂² − 2d₁d₂·cos(θ)).

By symmetric reasoning on triangle BCM (which has angle 180° − θ at M, so cos becomes −cos):

BC = ½√(d₁² + d₂² + 2d₁d₂·cos(θ))

The two formulas differ only in the sign of the cosine term. This gives the two adjacent sides of the parallelogram — opposite sides are equal (parallelogram property), so we get AB = CD and BC = AD.

Worked example 1 — basic case

Diagonals 10 and 14, angle between them 60°.

Side a = ½√(100 + 196 − 2·10·14·cos 60°) = ½√(296 − 140) = ½√156 = √39 ≈ 6.24.

Side b = ½√(100 + 196 + 140) = ½√436 ≈ 10.44.

Perimeter = 2(6.24 + 10.44) ≈ 33.36.

Worked example 2 — perpendicular diagonals

Diagonals 8 and 6, perpendicular (θ = 90°).

cos(90°) = 0, so the cosine term vanishes. Both formulas simplify:

Side a = Side b = ½√(64 + 36 + 0) = ½√100 = 5.

Both sides equal — so this is a rhombus (4 equal sides + perpendicular diagonals).

The 6-8-10 numbers hide a 3-4-5 right triangle inside.

Worked example 3 — diagonals equal length (rectangle)

Diagonals 13 and 13, angle 90° (diagonals equal AND perpendicular).

Side a = ½√(169 + 169 + 0) = ½√338 = ½√338 ≈ 9.19.
Side b = ½√(169 + 169 − 0) = ½√338 ≈ 9.19.

Wait — both equal? That would be a square. Let me try diagonals equal but not perpendicular.

Try: diagonals 13 and 13, angle 60°.

Side a = ½√(169 + 169 − 2·169·0.5) = ½√(338 − 169) = ½√169 = 6.5.
Side b = ½√(338 + 169) = ½√507 ≈ 11.27.

Unequal sides with equal diagonals → this is a rectangle (a parallelogram with equal diagonals).

How this connects to the parallelogram law

The parallelogram law says d₁² + d₂² = 2(a² + b²).

Summing the squared sides from our formulas:

a² + b² = ¼(d₁² + d₂² − 2d₁d₂cos θ) + ¼(d₁² + d₂² + 2d₁d₂cos θ) = ½(d₁² + d₂²)

Multiplying by 2: 2(a² + b²) = d₁² + d₂². ✓ Exactly the parallelogram law.

The angle between diagonals

The angle θ between the diagonals is a property of the parallelogram. Different parallelograms with the same diagonal lengths have different θ values, producing different sides:

• θ = 90° (perpendicular diagonals): the parallelogram is a rhombus.
• θ = 90° AND d₁ = d₂: a square.
• 0° < θ < 90°: a general parallelogram.
• θ = 180°: degenerate (the diagonals are along the same line — not a 2D parallelogram).

The reverse problem — finding diagonals from sides

If you know sides instead of diagonals, see the Parallelogram Theorem Calculator which gives diagonals from sides + angle.

The two directions are essentially inverse operations of each other. Both rely on the Law of Cosines applied to triangles formed by sides + diagonals.

Real-world applications

• Surveying. Measuring diagonals (often easier with long tape measures) to deduce side lengths.
• Manufacturing. Verifying that produced parallelogram-shaped parts have correct dimensions via diagonal checks.
• Aviation / engineering. Cross-bracing diagonals are sometimes the only practical measurements; sides follow from these formulas.
• Quality control. Squareness check via diagonal comparison — if diagonals are equal, parallelogram is a rectangle.

Common mistakes

• Confusing the angle at the intersection with one of the parallelogram's vertex angles. The angle θ in this formula is between the DIAGONALS at their intersection point, not at the parallelogram's corners.
• Forgetting the ½ in front of the square root. The formula divides by 2 (the bisection of each diagonal). Missing this gives sides twice as long.
• Using one formula for both sides. The two sides have different formulas (the cosine term has opposite signs). Compute each separately.
• Negative under the square root. This can happen if the inputs are physically inconsistent (e.g., diagonals that can't actually form a parallelogram). Check inputs.