Diagonals of Parallelogram: Formula, Examples

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What Are Diagonals of a Parallelogram?

Line segments connecting two non-adjacent vertices of a parallelogram are called “diagonals of a parallelogram.” 

A quadrilateral with opposite sides that are parallel and equal is known as a parallelogram. Its opposite angles are also equal. 

A parallelogram has two diagonals. The diagonals of a parallelogram connect the opposite vertices. 

A parallelogram and its two diagonals

Square, rectangle, rhombus are examples of a parallelogram.

Properties of Diagonals of a Parallelogram

  • Diagonals of a parallelogram bisect each other.
  • The diagonals of a square bisect each other at right angles.
  • The diagonals of a rectangle bisect each other, but not at right angles. 
  • The diagonals of a rhombus are perpendicular to each other.
  • Each diagonal divides the parallelogram in two congruent triangles.

Diagonals of Parallelogram Formula  

Let’s discuss two important formulas.

Finding Lengths of Diagonals of a Parallelogram

Diagonals of a parallelogram formulas

The above figure shows a parallelogram and its two diagonals. 

p & q are diagonals.

x & y are two adjacent sides of a parallelogram. 

$\angle \text{A}$ & $\angle\text{B}$ are the interior angles of a given parallelogram. 

How can we calculate the length of diagonals of a parallelogram?

The length of the diagonal of the parallelogram can be calculated by using the following formulas:

$p = \sqrt{x^{2} + y^{2}\;-\;2xy\; cos\;(A)} = \sqrt{x^{2} + y^{2} + 2xy\; cos\;(B)}$

$q = \sqrt{x^{2} + y^{2} + 2xy\; cos\;(A)} = \sqrt{x^{2} + y^{2} \;-\; 2xy\; cos\;(B)}$

If the measurement of two sides and one interior angle is given the above formula can be used for finding the length of diagonal of the parallelogram.

Relationship Between Sides and Diagonals of a Parallelogram

$p^{2} + q^{2} = 2 (x^{2} + y^{2})$

Here, Here, p & q are the diagonals of parallelogram

x & y are the adjacent sides of a parallelogram

If the measurement of two adjacent sides & one diagonal is given then the above formula can be used for finding the length of another diagonal of the parallelogram.

Fun Facts!

  • The diagonals of a parallelogram bisect each other at the point of intersection.
  • The length of the diagonals of a parallelogram is not equal.

Conclusion

This article gives a brief description of the diagonals of parallelograms, and how to calculate the length of diagonals of parallelograms. Let’s solve a few examples and practice problems.

Solved Examples on Diagonals of Parallelogram

1. Determine the length of diagonals of a parallelogram with side lengths 4 ft, 8 ft,  and angle $60^{\circ}$.

Finding the length of diagonals of a parallelogram

Solution: 

Here x $= 4$  ft & y $= 8$  ft

                      $\text{m} \angle \text{A} = 60^{\circ}$ 

Formula for calculating the length of diagonals is given as,

$p = \sqrt{x^{2} + y^{2}\;-\;2xy\; cosA}$

$= \sqrt{4^{2} + 8^{2} \;-\; 2(4)(8)\; cos(60^{\circ})}$

$= 6.92$ ft

$q = \sqrt{x^{2} + y^{2} + 2xy\; cosA}$

$ = \sqrt{4^{2} + 8^{2} + 2(4)(8)\; cos(60^{\circ})}$

$= 10.58$ ft

2. Determine the length of diagonals of a parallelogram with sides 3 inches and 6 inches, and the interior angle is 300.

Solution: 

Here,  x $= 3$ inches & y $= 6$ inches

              Also,   $\text{m}\;\angle\text{A} =30^{\circ}$

Formula for calculating the length of diagonals of the parallelogram is given as, 

$p = \sqrt{x^{2} + y^{2}\;-\;2xy\; cosA}$

$= \sqrt{3^{2} + 6^{2}\;-\;2(3)(6)\; cos\;30^{\circ}}$

$= 3.71$ inches

$q = \sqrt{x^{2} + y^{2} + 2xy\; cosA}$

$= \sqrt{3^{2} + 6^{2} + 2(3)(6)\; cos\;(30^{\circ})}$

$= 8.72$ inches

3. Determine the length of any one diagonal of a parallelogram if the length of the sides of a parallelogram and interior angle are 4 ft, 7 ft and 50o.

Solution: 

Given: Here,  x $= 4$ ft and y $= 7$  ft

              Also,    $\angle\text{A} = 50^{\circ}$

Formula for calculating the length of diagonals of the parallelogram is given as, 

$p = \sqrt{x^{2} + y^{2}\;-\;2xy\; cosA}$

$= \sqrt{4^{2} + 7^{2}\;-\;2(4)(7)\; cos(50^{\circ})}$

 $=  5.38$ ft

4. Determine the length of any one of the diagonals of a parallelogram having a length of sides 5ft, 7 ft, and one of the interior angles 450.

Solution: 

Here,  x $= 5$ ft & y $= 7$  ft

              Also,  $\angle\text{A} = 45^{\circ}$

Formula for calculating the length of diagonals of the parallelogram is given as, 

$p = \sqrt{x^{2} + y^{2}\;-\;2xy\; cos(A)}$

$= \sqrt{5^{2} + 7^{2}\;-\;2(5)(7)\; cos(45^{\circ})}$

$=  4.95$ ft

5. Determine the length of a diagonal of  a parallelogram with a side length of 5 ft and 8 ft if the length of another diagonal is 10 ft.

Solution:

Given: x $= 5$  ft,  y $= 8$  ft & p $= 10$  ft

As we know, the length of two sides and one diagonal is given for finding the length of another diagonal. We will use the formula of the relationship between the sides and diagonals of a parallelogram.

By using the formula,

$p^{2} + q^{2} = 2(x^{2} + y^{2})$

$\Rightarrow 10^{2} + q^{2} = 2(5^{2} + 8^{2})$

$\Rightarrow 100 + q^{2} = 2(25 + 64)$

$\Rightarrow 100 + q^{2} = 178$

$\Rightarrow q^{2} = 178\;-\;100$

$\Rightarrow q^{2} = 78$ 

By taking a square root,

$\Rightarrow q = 8.83$ ft

Practice Problems on Diagonals of Parallelogram

Diagonals of Parallelogram: Formula, Examples

Attend this quiz & Test your knowledge.

1

A parallelogram has _________ diagonals.

4
3
2
1
CorrectIncorrect
Correct answer is: 2
From the 4 vertices of a parallelogram, two diagonals can be drawn connecting two opposite corners.
2

Which of the following statements is not true for parallelogram?

Opposite sides are parallel.
Opposite angles are equal.
It has four vertices.
The length of the diagonal is equal.
CorrectIncorrect
Correct answer is: The length of the diagonal is equal.
The diagonals of the parallelogram are not equal. In the case of a rhombus, square, and rectangle, diagonals are equal.
3

Which of the following is not an example of a parallelogram?

Square
Rectangle
Kite
Rhombus
CorrectIncorrect
Correct answer is: Kite
Kite is not a parallelogram because opposite sides are not parallel and equal.
4

What is the length of the diagonal of a parallelogram with sides 3.5 inches, 6 inches, and interior angle $40^{\circ}$?

8.96 inches
7.5 inches
5.32 inches
4 inches
CorrectIncorrect
Correct answer is: 8.96 inches
Here, x $= 3.5$ in , y $= 6$ in & $\angle\text{A} = 40^{\circ}$
$p = \sqrt{x^{2} + y^{2}\;-\;2xy\; cosA}$
$= \sqrt{3.5^{2} + 6^{2}\;-\;2(3.5)(6)\; cos40^{\circ}}$
$= 8.96$ inches
5

What is the length of one of the diagonals if the length of the sides and one diagonal are 4.5 ft, 7 ft, 9 ft in respectively?

9.96 ft
10.5 ft
11.02 ft
12.5 ft
CorrectIncorrect
Correct answer is: 11.02 ft
As we know,
$p^{2} + q^{2} = 2(x^{2} + y^{2})$
$9^{2} + q^{2} = 2 (4.5^{2} + 9^{2})$
$81 + q^{2} = 202.5$
$q^{2} = 121.5$
$q=11.02$ ftin

Frequently Asked Questions on Diagonals of Parallelogram

Diagonals of a parallelogram bisect each other, but they are not equal.

A rectangle has equal diagonals which bisect each other and are perpendicular.

No, diagonals of a parallelogram bisect each other but not necessarily at $90^{\circ}$.

The parallelogram law states that the sum of the squares of the length of the four sides of a parallelogram is equal to the sum of the squares of the length of the two diagonals.

The number of diagonals of any quadrilateral can be calculated using the formula $\frac{n(n-3)}{2}$, where n is the number of sides of a given polygon.