If two diagonals of a rhombus are 20cm and 21cm find the side and perimeter of the rhombus

The perimeter of a rhombus is the sum of all its sides. Rhombus is a quadrilateral in which all four sides are of the same measure. A rhombus is always a parallelogram but a parallelogram may not necessarily be a rhombus always. Let us study more about the perimeter of rhombus in this article.

What is the Perimeter of a Rhombus?

The perimeter of a rhombus is the total measure of its boundary and it is calculated by adding the length of all its sides. Since all the four sides of a rhombus are equal, the basic formula to find the perimeter of a rhombus is: Perimeter = 4a; where 'a' is the side of the rhombus. The perimeter of a rhombus is expressed in linear units like inches, yards, centimeters, and so on.

Properties of a Rhombus

There are some basic properties that help us to identify a rhombus. Observe the following rhombus ABCD to relate to its properties given below.

We can identify and distinguish a rhombus with the help of the following properties:

• All four sides are equal.
• The opposite sides are parallel.
• The opposite angles are equal.
• The sum of any two adjacent angles is 180o.
• The diagonals bisect each other at right angles.
• Each diagonal bisects the vertex angles.

Now, let us read about the formulas that are used to find the perimeter of a rhombus in different cases. The formulas differ according to the known dimensions.

Perimeter of Rhombus Formula With Sides

As discussed earlier, the perimeter of a rhombus is the sum of the lengths of all its sides. We know that the sides of a rhombus are of equal lengths. Let us consider a rhombus of side length 'a'. Then, the perimeter of the rhombus is a + a + a + a which is 4a. Thus, the perimeter of the rhombus formula is: P = 4a.

Example: Find the perimeter of a rhombus that has a side length of 10 units.

Solution:

The side length of the given rhombus, a = 10 units.

Its perimeter = 4a = 4 × 10 = 40 units.

Perimeter of Rhombus Formula With Diagonals

When the length of the diagonals of a rhombus is known, we find the side length of the rhombus using the Pythagoras theorem. Here, we make use of the following properties of a rhombus:

• A rhombus is divided into 4 congruent right-angled triangles by its two diagonals.
• The diagonals bisect each other at right angles.

Let us consider a rhombus ABCD with diagonals p and q and with side length 'a'.

Let us take triangle AOD. Since the diagonals bisect at right angles, AO can be written as p/2 and OD can be written as q/2. Now, if we apply the Pythagoras theorem for triangle AOD, we get

\(a^2 = \dfrac{p^2}{4}+\dfrac{q^2}{4}\)

\(a =\dfrac{ \sqrt{p^2+q^2}}{2}\)

Since we know that Perimeter of the rhombus = 4a

Let us substitute the value of : \(a =\dfrac{ \sqrt{p^2+q^2}}{2}\) in the formula: P = 4a

Perimeter of rhombus = 4 × \(\dfrac{ \sqrt{p^2+q^2}}{2}\) (or)

Perimeter of rhombus = \(2\sqrt{p^2+q^2}\)

Note: We do not need to remember this formula. We can use the Pythagoras theorem to find the side length of the rhombus using the diagonals and then we can apply the perimeter of rhombus formula to be, P = 4 × side length.

1. Example 1: Find the perimeter of a rhombus with diagonals 8 inches and 6 inches respectively.

   Solution:

   Method 1:

   The lengths of the diagonals of the given rhombus are, p = 8 inches and q = 6 inches.

   Using the perimeter of rhombus formula using diagonals,

   Perimeter = 2 \(\sqrt{p^2+q^2}\)

   Perimeter = 2 \(\sqrt{8^2+6^2}\) = 2 \(\sqrt{100}\) = 2 × 10 = 20 inches.

   Method 2:

   Let us assume that the side length of the rhombus is 'a'.

   Since the diagonals bisect each other at right angles,

   

   By Pythagoras theorem,

   a2 = 32 + 42 = 25

   a = 5 inches.

   Thus, the perimeter of the rhombus is, 4a = 4 × 5 = 20 inches.

   Answer: The perimeter of the given rhombus = 20 inches.

2. Example 2: ABCD is a rhombus with one side as 7 units. Find the perimeter of ABCD.

   Solution:

   Let us assume that the side length of the rhombus ABCD is 'a' units.

   The side length of the given rhombus, a = 7 units.

   We know that perimeter of a rhombus with sides = 4a = 4 × 7 = 28 units.

   Answer: The perimeter of the rhombus = 28 units.

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FAQs on Perimeter of Rhombus

The perimeter of a rhombus is the sum of all its sides. Since all the four sides of a rhombus are equal, the formula used to find the perimeter of a rhombus is: Perimeter = a + a + a + a = 4a, where 'a' represents the side length of the rhombus.

How to Find the Perimeter of Rhombus When only Diagonals are Given?

When the diagonals of a rhombus are known, then the formula to find its perimeter is, P = 2\(\sqrt{p^2+q^2}\), where 'p' and 'q' are the diagonals. There is another way to find the perimeter with the help of diagonals. We know that both the diagonals of a rhombus divide it into 4 congruent right-angled triangles, so we apply the Pythagoras theorem to one of the triangles to find the side length 'a' of the rhombus, and then we find the perimeter using the formula: P = 4a.

How to Find the Perimeter of Rhombus?

There are different ways to find the perimeter of a rhombus which depend on the given dimensions.

• When one side of a rhombus is known, the perimeter = 4a; where 'a' is the side length.
• When the two diagonals of a rhombus are known, the perimeter = 2 \(\sqrt{p^2+q^2}\); where 'p' and 'q' are the diagonals.

What is the Area and Perimeter of a Rhombus?

The area of a rhombus is the space occupied by it. This is calculated with the help of different formulas which depend on the type of dimensions given and is expressed in square units.

• When the base and height of a rhombus are known, then the area of rhombus = base × height.
• When the diagonals of a rhombus are known, then the area = (diagonal 1 × diagonal 2)/2.

The perimeter of a rhombus is the total length of its boundary. This is calculated with the help of different formulas which depend on the given dimensions and is expressed in linear units.

• When one side of a rhombus is known, the perimeter = 4 × side
• When the two diagonals of a rhombus are known, the perimeter = 2 \(\sqrt{p^2+q^2}\); where 'p' and 'q' are the diagonals.

What is the Perimeter of a Rhombus Formula When Side Length is Given?

Since all the 4 sides of a rhombus are equal, its perimeter is obtained by multiplying its side by 4. The formula to calculate the perimeter of a rhombus with side 'a' is: P = 4a.

How to Find the Side Length When the Perimeter of a Rhombus is Given?

We know that the perimeter of a rhombus with side length 'a' is calculated with the help of the formula, P = 4a. Thus, the side length of the rhombus can be obtained by dividing its perimeter by 4.

What is the Perimeter of a Rhombus with Diagonals 16 and 30?

When the diagonals of a rhombus are known, we use the formula: perimeter = 2 \(\sqrt{p^2+q^2}\); where 'p' and 'q' are the diagonals. Substituting the given values 16 and 30 in the formula, P = 2 \(\sqrt{16^2+30^2}\), we get the perimeter as 68 units.

Diagonals of rhombus are perpendicular bisector of each other.

Hence side of the rhombus =

Perimeter = 4×14.5 = 58 cm

Answered by Expert 18th December 2018, 12:22 PM

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Diagonals of a rhombus are 20 cm and 21 cm respectively, then find the side of rhombus and its perimeter.

Let ABCD be the rhombus.AC = 20 cm and BD = 21 cmWe know that the diagonals of a rhombus bisect at right angles.So, AO = OC = 10 cm

And BO = OD = 10.5 cm

In ΔAOB,

`AO^2 + OB^2 = AB^2`

`=> 10^2 + 10.5 ^2 = AB^2`

`=> 100 + 110.25 = AB^2`

`=> AB^2 = 210.25`

`=> AB = 14.5  "cm" `

Thus, each side of rhombus = 14.5 cmPerimeter = AB + BC + CD + DA = 14.5 + 14.5 + 14.5 + 14.5

= 58 cm.

Concept: Types of Quadrilaterals - Properties of Rhombus

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