# 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

1. A parallelogram has adjacent sides 7 cm and 10 cm with the included angle 45°. Find both diagonals.

2. A parallelogram has sides 8 cm and 6 cm, and one diagonal of length 12 cm. Find the other diagonal using the parallelogram law.

3. 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.