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Diagonal of Parallelogram
The diagonal of a parallelogram is the line segment that connects its nonadjacent vertices. A parallelogram has 2 diagonals and the length of the diagonals of a parallelogram can be found by using various formulas depending on the given parameters and dimensions. Let us learn more about the diagonals of a parallelogram in this article.
1.  What is the Diagonal of a Parallelogram? 
2.  Diagonal of Parallelogram Formula 
3.  FAQs on Diagonal of Parallelogram 
What is the Diagonal of Parallelogram?
The diagonals of a parallelogram can be drawn by joining the two nonadjacent vertices of the parallelogram. It should be noted that the 2 diagonals of a parallelogram bisect each other and they divide the parallelogram into congruent triangles.
Diagonal of Parallelogram Formula
The formula for the diagonals of a parallelogram is used to calculate the length of the diagonals of a given parallelogram. There are different formulas for different kinds of parallelograms. Observe the figure given below which shows a parallelogram along with its diagonals. Here 'p' and 'q' are the diagonals and 'x' and 'y' are the two sides of the parallelogram.
The simple formula for finding the length of the diagonals of a parallelogram is given below. For this formula, we need the length of the sides and any of the known angles. If we follow the figure given above, we can observe that:
 p and q are taken to be the length of the diagonals respectively.
 x and y are the sides of the parallelogram.
 Angle A and Angle B are two interior angles of the parallelogram.
Formula 1: For any parallelogram, the formula for the length of the diagonals is expressed as:
\(p = \sqrt{x^2 + y^2  2xy \cos A} = \sqrt{x^2 + y^2 + 2xy \cos B} \)
\(q = \sqrt{x^2 + y^2 + 2xy \cos A} = \sqrt{x^2 + y^2  2xy \cos B} \)
Formula 2: Another formula which expresses the relationship between the length of the diagonals and sides of the parallelogram is:
p^{2} + q^{2 }= 2(x^{2 }+ y^{2})
Where,
 p and q are the diagonals respectively.
 x and y are the sides of the parallelogram.
It should be noted that a square, a rectangle, and a rhombus come under the category of parallelograms. And since they have different properties, the formula that is used to find their diagonals is also different. For example, the diagonal of a square (d) = a√2; where 'd' is the diagonal and 'a' is the side of the square. The diagonal of a rectangle (d) = √( l^{2} + w^{2}), where l = length of the rectangle and w = width of the rectangle. Therefore, the formula for the diagonal of a parallelogram varies for different kinds of parallelograms.
Properties of Diagonal of Parallelogram
The following points show the properties of the diagonals of a parallelogram. Since a parallelogram includes a square, a rectangle, a rhombus, the diagonals of these figures have a few common properties and a few different ones.
 The diagonals of a parallelogram always bisect each other.
 In a square, the diagonals are equal and bisect each other at right angles.
 In a rectangle, the diagonals are equal and they bisect each other but not at right angles.
 In a rhombus, the diagonals may not be necessarily equal, but they are perpendicular to each other.
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Examples using Diagonal of Parallelogram Formula

Example 1: Find the length of the diagonals of the rhombus of side length 4 inches, 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: 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: The length of the diagonal is 58 ft.

Example 3: Calculate the length of the diagonal of a parallelogram with sides 4 units, 6 units and an interior angle A which is equal to 60 degrees.
Solution:
Given, a = 4 units, b = 6 units, angle A = 60°
Using diagonal of parallelogram formula,
\(p = \sqrt{x^2 + y^2  2xy \cos A}\)
Putting the values in the formula for p:
\(p = \sqrt{4^2 + 6^2  24}\)= \(\sqrt28\) = 5.291Answer: Diagonal of parallelogram = 5.291 units.
FAQs on Diagonal of Parallelogram
What is the Diagonal of a Parallelogram?
The diagonal of a parallelogram is the line segment that joins the two nonadjacent vertices of the parallelogram. It is to be noted that 2 diagonals can be drawn in a parallelogram.
What is the Diagonal of a Parallelogram Formula?
A simple formula which is used to find the length of the diagonals of a parallelogram needs the value of the interior angles and the length of the sides. For any parallelogram, the formula for the length of the diagonals is expressed as, \( p = \sqrt{x^2 + y^2  2xy \cos A} = \sqrt{x^2 + y^2 + 2xy \cos B} \) and \( q = \sqrt{x^2 + y^2 + 2xy \cos A} = \sqrt{x^2 + y^2  2xy \cos B} \), where p and q are the lengths of the diagonals, angle A and angle B are the given interior angles and x and y are the sides of the parallelogram.
How to Use the Diagonal of a Parallelogram Formula?
For any parallelogram, let p and q be the lengths of the diagonals and x and y be the sides of the parallelogram then
 Step 1: Check for the given parameters, the values of the sides of the parallelograms, and the corresponding angles.
 Step 2: Substitute the values in the formula, \( p = \sqrt{x^2 + y^2  2xy \cos A} = \sqrt{x^2 + y^2 + 2xy \cos B} \) and \( q = \sqrt{x^2 + y^2 + 2xy \cos A} = \sqrt{x^2 + y^2  2xy \cos B} \).
What are the Components of the Diagonal of Parallelogram Formula?
The formula for the diagonal of parallelogram helps to find the length of the diagonals by using the length of the sides and any of the known angles. Thus, its components include the sides of the parallelogram and the corresponding angles.
Do the Diagonals of a Parallelogram Bisect Each Other?
Yes, the diagonals of a parallelogram bisect each other. This means that the diagonals of a parallelogram divide each other into 2 equal parts.
Are the Diagonals of a Parallelogram Equal?
A parallelogram includes a square, a rectangle, and a rhombus. While the diagonals of a square and rectangle are equal, the diagonals of a rhombus may not be necessarily equal.
How to Find the Diagonals of a Parallelogram without Angles?
The length of the diagonals of a parallelogram can be calculated even when the interior angles are not given. For example, if the parallelogram is a rectangle, we know that the diagonals of a rectangle form a rightangled triangle. So, in this case, if the sides of the rectangle are known, the length of the diagonal can be calculated using the Pythagoras theorem because the diagonal becomes the hypotenuse. This method can also be applied if the given parallelogram is a square.
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