Diagonal of Parallelogram Formula
Diagonals of a parallelogram formula are used to find the length of the diagonals of a parallelogram. The diagonals of a parallelogram are the connecting line segments between opposite vertices of the parallelogram. Using this formula we can find out the lengths of the diagonals only using the length of the sides and any of the known angles. Let us see more about the diagonals of a parallelogram along with the solved examples in the next section.
What is the Diagonal of Parallelogram Formula?
For any parallelogram abcd, the formula for the length of the diagonals is:
\[ p = \sqrt{x^2 + y^2  2xy \cos A} = \sqrt{x^2 + y^2 + 2xy \cos B} \]
\[ q = \sqrt{x^2 + y^2 + 2xy \cos A} = \sqrt{x^2 + y^2  2xy \cos B} \]
Where, p and q are the length of the diagionals respectively.
One more special formula relating the lengths of the diagonals and sides of the parallelogram is:
\[p^2 + q^2 = 2(a^2 + b^2)\]
Let's take a quick look at a couple of examples to understand diagonals of parallelogram formula, better.
Solved Examples using Diagonal of Parallelogram Formula

Example 1:
Find the length of the diagonals of the rhombus of side length 4in, if the interior angles are 120°, and 60°.
Solution:
Given, Interior angle A = 120°, and angle B = 60°.
x = 4, y = 4Using diagonal of parallelogram formula,
\[ p = \sqrt{x^2 + y^2  2xy \cos A} \]
\[ q = \sqrt{x^2 + y^2 + 2xy \cos A}\]
Putting the values in the formula for p:
\[\begin{align} p &= \sqrt{4^2 + 4^2  2 \times 4 \times 4 \times \cos 60} \\ &= \sqrt{32  16} \\ p &= 4 \end{align}\]
Now, doing same for q,
\[\begin{align} q &= \sqrt{4^2 + 4^2 + 2 \times 4 \times 4 \times \cos 60} \\ &= \sqrt{32 + 16} \\ p &= \sqrt {48} \\ p &= 6.92 \end{align}\]Answer: Hence the length of the diagonals are 4 in and 6.92 in.

Example 2: For a parallelogram ABCD, if the length of the adjacent sides is 35 ft and 82 ft. If one of the interior angles is 37°. Find the length of any diagonal.
Solution:
Given, Interior angle A = 37°
x = 35 ft, y = 82 ftUsing diagonal of parallelogram formula,
\[ p = \sqrt{x^2 + y^2  2xy \cos A} \]
Putting the values in the formula for p:
\[\begin{align} p &= \sqrt{35^2 + 82^2  2 \times 35 \times 82 \times \cos 37} \\ &= \sqrt{3365} \\ p &= 58 \end{align}\]Answer: Hence the length of the diagonal is 58 ft.