Diagonal of Parallelogram Formula: The name parallelogram comes from the Greek word parallelogrammon, which means bounded by parallel lines. As a result, a parallelogram is a quadrilateral with parallel lines on opposite sides. The opposite sides of a parallelogram will be parallel and equal. The diagonals of a parallelogram are the line segments that connect the parallelogram’s opposite vertices.

We can calculate the lengths of the diagonals of a parallelogram if we know the measure of its adjacent sides and the adjacent angles. In this article, we will discuss the diagonal of the parallelogram formula in detail.

What is a Parallelogram?

A parallelogram is a type of quadrilateral formed by parallel lines. The opposite sides of a parallelogram are parallel and congruent. A parallelogram’s angle between consecutive sides can vary, but the opposite angles are equal.

Examine the diagram below, which represents the three kinds of parallelograms. These are special cases of parallelograms.

Diagonal

A diagonal is a line segment that connects two corners of a polygon that is not an edge. As a result, we may make a diagonal by connecting any two corners (vertices) that are not previously connected by an edge.

Number of Diagonals of a Parallelogram

For an \(n\) -sided regular polygon, the number of diagonals can be obtained using the formula given below:
Number of diagonals \( = \frac{{n\left( {n – 3} \right)}}{2}\)
For a parallelogram, \(n = 4\)
Therefore, the number of diagonals in a parallelogram \( = \frac{{4\left( {4 – 3} \right)}}{2} = 2.\)

Diagonals of a Parallelogram

The diagonals of a parallelogram are the line segments joining the opposite vertices of the parallelogram.
A parallelogram has two pairs of opposite vertices, and hence it has two diagonals.

In the parallelogram \(ABCD,\,AC\) and \(BD\) are the diagonals.

Diagonals of Parallelogram Formula

The diagonals of a parallelogram formula are used to determine the length of a parallelogram’s diagonals. Using the lengths of the sides and the measure of the angles, we can calculate the diagonal lengths.

For any parallelogram \(ABCD,\) let \(a\) and \(b\) be the length of sides of the parallelogram and \(p\) and \(q\) be the lengths of the diagonals then,
Using the cosine rule on angle \(A\) and \(B,\) we get
\(p = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,\left( A \right)} = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,\left( B \right)} \)
\(q = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,\left( A \right)} = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,\left( B \right)} \)

Relationship Between Sides and Diagonals of a Parallelogram

One more special formula related to the lengths of the diagonals and sides of the parallelogram is given by:
\({p^2} + {q^2} = 2\left( {{a^2} + {b^2}} \right)\)
Where \(p\) and \(q\) are the lengths of the diagonals, respectively, and \(a\) and \(b\) are the sides of the parallelogram.

Types of Parallelogram

A parallelogram may be classified into several kinds based on its various properties. It is mostly classified into three distinct categories:
1. Rectangle
2. Square
3. Rhombus

1. Rectangle

A rectangle is a parallelogram with four right angles and two sets of equal and parallel opposite sides.

Diagonals of a Rectangle

A rectangle’s diagonal is a line segment connecting any two of its non-adjacent vertices. The diagonals of the following rectangle are \(AC\) and \(BD.\) As you can see, the lengths of \(AC\) and \(BD\) are equal. A diagonal divides a rectangle into two right triangles, each having sides equal to the rectangle’s sides and a hypotenuse which is the diagonal of the triangle.

The length of a diagonal \(d\) of a rectangle whose length is \(l\) units and breadth is \(b\) units is calculated by the Pythagoras theorem.
Using Pythagoras theorem, we get:
\({d^2} = {l^2} + {b^2}\)
Therefore, length of diagonal of a rectangle \( = \sqrt {{l^2} + {b^2}} .\)

2. Square

A parallelogram with four equal sides and four right angles is known as a square.

Diagonals of a Square

A square’s diagonal is a line segment connecting any two of its opposite vertices.

In the given square, the lengths of the line segments \(AC\) and \(BD\) are the same. Any square’s diagonal divides it into two equal right triangles, with the diagonal forming the hypotenuse of the right triangles.
The Pythagoras theorem is used to compute the length of a diagonal \(d\) of a square with side length \(a\) units.

According to Pythagoras’ theorem,
Length of diagonal of a square, \(d = \sqrt {{a^2} + {a^2}} = \sqrt {2{a^2}} = a\sqrt 2 \,{\rm{units}}\)

3. Rhombus

A rhombus is a parallelogram with four equal sides and equal opposite angles.

Diagonals of a Rhombus

The diagonals of a rhombus are the line segments that connect the opposite vertices and bisect them at a \({90^{\rm{o}}}\) angle, ensuring that the two halves of each diagonal are equal in length. A rhombus is a diamond-shaped quadrilateral with equal sides on all four sides. Unless the rhombus is a square, the diagonals of a rhombus will have distinct values.
For a rhombus \(ABCD\)
\(AB = BC = CD = AD = a\)
\(AC = p,\,BD = q\)
\(\theta = \)angle at vertex \(A\) then,
\(p = \sqrt {{a^2} + {a^2} + 2\,{a^2}\,\cos \,\theta } = \sqrt {2\,{a^2} + 2\,{a^2}\,\cos \,\theta } = a\sqrt {2 + 2\,\cos \,\theta } \)
\(q = \sqrt {{a^2} + {a^2} = 2\,{a^2}\,\cos \,\theta } = \sqrt {2\,{a^2} – 2\,{a^2}\,\cos \,\theta } = a\sqrt {2 – 2\,\cos \,\theta } \)

Properties of Diagonal of a Parallelogram

The diagonals of the parallelogram \(ABCD\) are \(AC\) and \(BD.\) The properties of a parallelogram’s diagonals are as follows:
1. Diagonals of a parallelogram bisect each other. \(OB = OD\) and \(OA = OC\)
2. Each diagonal divides the parallelogram into two congruent triangles.
3. Parallelogram Law: The sum of the squares of the sides is equal to the sum of the squares of the diagonals.
\(A{B^2} + B{C^2} + C{D^2} + A{D^2} = A{C^2} + B{D^2}\)

Solved Examples – Diagonal of Parallelogram Formula

Q.1. Explain how a square is:
(i) a quadrilateral
(ii) a parallelogram
(iii) a rhombus
(iv) a rectangle
Ans: (i) A square is a quadrilateral since it is a closed two-dimensional shape with four straight line segments.
(ii) A square is a parallelogram since it has both pairs of opposite sides parallel and equal.
(iii) A square is a rhombus since it has four equal sides and diagonals bisects each other at \({90^{\rm{o}}}.\)
(iv) A square is a rectangle since it has each adjacent angle a right angle and opposite sides are equal.

Q.2. Find the length of the diagonals of the rhombus of side length 4 cm, if the interior angles are \({120^{\rm{o}}}\) and \({60^{\rm{o}}}.\)
Ans: Given, Interior \(\angle A = {120^{\rm{o}}}\) and \(\angle B = {60^{\rm{o}}}.\)
\(a = 4\,{\rm{cm}}\)
Using diagonal of rhombus formula,
\(p = a\sqrt {2 + 2\,\cos \,A} \)
\(q = a\sqrt {2 – 2\,\cos \,A} \)
Putting the values in the formula for \(p:\)
\(p = 4\sqrt {2 + 2\,\cos \,{{120}^{\rm{o}}}} \)
\(p = 4\sqrt {2 + 2 \times \left( { – \frac{1}{2}} \right)} \)
\(p = 4\sqrt {2 – 1} = 4\,{\rm{cm}}\)
Similarly, for\(q,\)
\(q = 4\sqrt {2 – 2\,\cos \,{{120}^{\rm{o}}}} \)
\(q = 4\sqrt {2 – 2 \times \left( { – \frac{1}{2}} \right)} \)
\(q = 4\sqrt {2 + 1} = 4 \times 1.732 = 6.928\,{\rm{cm}}\)
Hence, the length of the diagonals is \(4\,{\rm{cm}}\) and \(6.928\,{\rm{cm}}{\rm{.}}\)

Q.3. Find the length of diagonals of a rectangle whose length is 3 cm and breadth is 4 cm?
Ans: Given, \(l = 3\,{\rm{cm}}\) and \(b= 4\,{\rm{cm}}\)
Using diagonal of rectangle formula,
\(d = \sqrt {{l^2} + {b^2}} \)
\(d = \sqrt {{3^2} + {4^2}} \)
\(d = \sqrt {9 + 16} = \sqrt {25} = 5\,{\rm{cm}}\)
Hence, the length of the diagonal of the given rectangle is \(5\,{\rm{cm}}.\)

Q.4. Name the quadrilaterals whose diagonals:
(i) bisect each other
(ii) are perpendicular bisectors of each other
(iii) are equal
Ans: (i) The diagonals bisect each other in a rhombus, parallelogram, rectangle, or square.
(ii) The quadrilaterals, which have diagonals as perpendicular bisectors, are rhombus and square.
(iii) A square or rectangle is formed when the diagonals are equal.

Q.5. Find the diagonal of a parallelogram with sides 2 cm, 6 cm, and angle \({45^{\rm{o}}}\)?
Ans: Given \(a = 2\,{\rm{cm}},\,b = 6\,{\rm{cm}}\) and \(\angle A = {45^{\rm{o}}}\)
The formula of diagonal is,
\(p = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,\left( A \right)} \)
\(p = \sqrt {{2^2} + {6^2} – 2 \times 2 \times 6 \times \cos \,\left( {{{45}^{\rm{o}}}} \right)} \)
\(p = \sqrt {4 + 36 – 24 \times 0.707} \)
\(p = \sqrt {40 – 16.968} \)
\(p = \sqrt {23.032} \)
\(p = 4.799\,{\rm{cm}}\)
Hence, the length of the diagonal of the parallelogram is \(4.799\,{\rm{cm}}{\rm{.}}\)

Summary

This article discussed that a parallelogram is a polygon whose opposite sides are equal and parallel. In this article, we also learnt about the definition of a parallelogram, its diagonal, types of a parallelogram and their length of diagonals, properties of a parallelogram, formula to find the length of the diagonals of a parallelogram. We also solved some examples concerning all these.

Frequently Asked Questions (FAQ) – Diagonal of Parallelogram Formula

Q.1. Is the diagonal of the parallelogram equal?
Ans: No, the diagonals of a parallelogram are not equal. But, the diagonals divide the parallelogram into two pairs of congruent triangles. The diagonals will be equal if a parallelogram is a rectangle or square.

Q.2. What is the diagonal of a parallelogram?
Ans: The diagonals of a parallelogram are the line segments joining the opposite vertices of the parallelogram. There are two diagonals in a parallelogram.

Q.3. What is the diagonal of a \(5\) inches square?
Ans: We know that the length of diagonal of a square is \(a\sqrt 2 ,\) where \(a\) is the side of the square.
Therefore, for a \(5\,{\rm{inches}}\) square, the length of the diagonal is \(5\sqrt 2 \,{\rm{inches}}{\rm{.}}\)

Q.4. What is the diagonal formula?
Ans: The diagonals of a parallelogram formula is used to determine the length of a parallelogram’s diagonals. Using the lengths of the sides and the measure of angles, we can find the length of the diagonals of a parallelogram.
For any parallelogram, let \(a\) and \(b\) be the sides of the parallelogram and \(p\) and \(q\) be the lengths of the diagonals then.
\(p = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,\left( A \right)} = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,\left( B \right)} \)
\(q = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,\left( A \right)} = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,\left( B \right)} \)

Q.5. How to use the diagonal of a parallelogram formula?
Ans: For any parallelogram, let \(a\) and \(b\) be the sides of the parallelogram and \(p,\) and \(q\) be the lengths of the diagonals then
Step 1: Check for the given parameters, the sides of the parallelograms, and the corresponding angles.
Step 2: Put the values in the formula for \(a,\,b,\,A\) and \(B\) and solve to get the values of \(p\) and \(q\) which are the length of the diagonals of the parallelogram.
\(p = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,\left( A \right)} = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,\left( B \right)} \)
\(q = \sqrt {{a^2} + {b^2} – 2\,ab\,\cos \,\left( A \right)} = \sqrt {{a^2} + {b^2} + 2\,ab\,\cos \,\left( B \right)} \)

Now you are provided with all the necessary information on the diagonal of a parallelogram formula and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.