Diagonal of Parallelogram Formula — p = √(x² + y² − 2xy·cos A)

TL;DR

The diagonal of parallelogram formula computes a diagonal's length from the two adjacent sides and the included angle: p=√(x² + y² − 2xy·cos A). This article gives the formula, derives it from the law of cosines, gives the companion parallelogram law p² + q² = 2(x² + y²), and walks through three worked examples and the most common student mistake.

A Diagonal Is Just the Law of Cosines in Disguise

Most students meet the parallelogram in Class 8 — opposite sides equal, opposite angles equal, diagonals bisect each other. Then in Class 10 or 11 they're asked to compute the diagonal length, and the formula looks intimidating: p=√(x² + y² − 2xy·cos A). Square root, cosine, two squared sides — it reads like an equation pulled from advanced geometry.

The diagonal of parallelogram formula says:

p=√(x² + y² − 2xy·cos A),

where x and y are adjacent sides and A is the angle between them.

The whole formula is one law-of-cosines step.

The Diagonal of Parallelogram Formula

For a parallelogram with adjacent sides x and y and the angle A between them, the two diagonals are:

p=√(x² + y² − 2xy·cos A) and q=√(x² + y² + 2xy·cos A).

Notice the sign — the shorter diagonal carries the minus sign (opposite vertex angle is acute), the longer diagonal carries the plus. The two diagonals split the included angle differently and that splits the cosine sign.

The companion identity — the parallelogram law:

p² + q² = 2(x² + y²).

The sum of the squares of the two diagonals equals twice the sum of the squares of the two sides.

How the Diagonal of Parallelogram Formula Is Derived — One Application of the Law of Cosines

Take a parallelogram ABCD with AB=x along the base, AD=y rising at angle A, and diagonal AC=p from vertex A to opposite vertex C.

The diagonal AC closes a triangle with two of the parallelogram's sides — but not directly with x and y. The triangle is A, B, C, where:

AB=x
BC=y (since opposite sides of a parallelogram are equal, BC=AD=y)
The angle at B — let's call it B — is supplementary to angle A, since consecutive angles of a parallelogram sum to 180°.

The law of cosines applied to triangle ABC gives:

AC² = AB² + BC² − 2·AB·BC·cos(B).

Using the identity cos(180°−A) = −cos(A):
p² = x² + y² − 2xy·(−cos A) = x² + y² + 2xy·cos A.

Actually, the convention depends on which diagonal is which. The diagonal from the acute-angle vertex picks up the minus sign (the formula in the boxed statement); the diagonal from the obtuse-angle vertex picks up the plus sign.

Three Worked Examples of Diagonal of Parallelogram Formula

Quick. A parallelogram has adjacent sides 6 cm and 8 cm with the included angle 60°. Find the shorter diagonal.

Use p=√(x² + y² − 2xy·cos A) with x=6, y=8, A=60°, cos(60°)=0.5:

p=√(36 + 64 − 2·6·8·0.5) = √(100 − 48) = √(52) ≈ 7.21 cm.

Standard (Wrong Path First). A parallelogram has sides 5 cm and 12 cm with one angle 120°. Find both diagonals.

The wrong path:
p=√(25 + 144 − 2·5·12·0.5) = √(169 − 60) = √(109) ≈ 10.44 cm.

The correct approach uses cos(120°) = −0.5:

p=√(25 + 144 − 2·5·12·(−0.5)) = √(169 + 60) = √(229) ≈ 15.13 cm.

Take These for a Test Drive — Three Problems

A parallelogram has adjacent sides 7 cm and 10 cm with the included angle 45°. Find both diagonals.
A parallelogram has sides 8 cm and 6 cm, and one diagonal of length 12 cm. Find the other diagonal using the parallelogram law.
A square has diagonal 525 cm. Find its side length, then verify the parallelogram law.

Frequently Asked Questions

What is the diagonal of parallelogram formula?
p = √(x² + y² − 2xy·cos A) for one diagonal, q = √(x² + y² + 2xy·cos A) for the other, where x and y are adjacent sides and A is the included angle.

How do you find both diagonals of a parallelogram?
Use the formula with cos(A) for the diagonal opposite angle A, and cos(180°−A) for the other. Or use the parallelogram law p² + q² = 2(x² + y²) to get the second diagonal once you have the first.