Mathwords logoMathwords

Parallelogram Law — Definition, Formula & Examples

The Parallelogram Law states that the sum of the squares of all four sides of a parallelogram equals the sum of the squares of its two diagonals.

For a parallelogram with sides of length aa and bb and diagonals of length d1d_1 and d2d_2, the identity d12+d22=2a2+2b2d_1^2 + d_2^2 = 2a^2 + 2b^2 holds. Equivalently, in vector form, ∥u+v∥2+∥u−v∥2=2∥u∥2+2∥v∥2\|\mathbf{u} + \mathbf{v}\|^2 + \|\mathbf{u} - \mathbf{v}\|^2 = 2\|\mathbf{u}\|^2 + 2\|\mathbf{v}\|^2 for any vectors u\mathbf{u} and v\mathbf{v}.

Key Formula

d12+d22=2a2+2b2d_1^2 + d_2^2 = 2a^2 + 2b^2
Where:
  • aa = Length of one pair of parallel sides
  • bb = Length of the other pair of parallel sides
  • d1d_1 = Length of one diagonal
  • d2d_2 = Length of the other diagonal

How It Works

The two diagonals of a parallelogram are formed by adding and subtracting the side vectors. When you know any three of the four quantities — two side lengths and two diagonal lengths — you can solve for the missing one. In the vector interpretation, u+v\mathbf{u} + \mathbf{v} gives one diagonal and u−v\mathbf{u} - \mathbf{v} gives the other, so squaring and adding eliminates the cross terms involving the angle between the sides.

Worked Example

Problem: A parallelogram has sides of length 5 and 12. One diagonal measures 13. Find the length of the other diagonal.
Write the formula: Apply the Parallelogram Law.
d12+d22=2(5)2+2(12)2d_1^2 + d_2^2 = 2(5)^2 + 2(12)^2
Compute the right side: Square each side length and double.
2(25)+2(144)=50+288=3382(25) + 2(144) = 50 + 288 = 338
Substitute the known diagonal: Plug in d1=13d_1 = 13 and solve for d2d_2.
169+d22=338  ⟹  d22=169  ⟹  d2=13169 + d_2^2 = 338 \implies d_2^2 = 169 \implies d_2 = 13
Answer: The other diagonal is also 13 units long. (This parallelogram happens to be a rectangle.)

Why It Matters

The Parallelogram Law provides a quick way to find a missing diagonal or side length without needing an angle measurement. In physics and engineering, the vector form is used to verify inner-product space properties and to decompose forces acting along different directions.

Common Mistakes

Mistake: Writing d12+d22=a2+b2d_1^2 + d_2^2 = a^2 + b^2 without the factor of 2 on the right side.
Correction: A parallelogram has two sides of length aa and two of length bb, so the correct right side is 2a2+2b22a^2 + 2b^2, accounting for all four sides.