How to find the length of diagonal of a rhombus? (original) (raw)

Last Updated : 18 Feb, 2024

Rhombus is also known as a four-sided quadrilateral. It is considered to be a special case of a parallelogram. A rhombus contains parallel opposite sides and equal opposite angles. A rhombus is also known by the name diamond or rhombus diamond. A rhombus contains all the sides of a rhombus as equal in length. Also, the diagonals of a rhombus bisect each other at right angles.

**Properties of a Rhombus

A rhombus contains the following properties:

• A rhombus contains all equal sides.
• Diagonals of a rhombus bisect each other at right angles.
• The opposite sides of a rhombus are parallel in nature.
• The sum of two adjacent angles of a rhombus is equal to 180o.
• There is no inscribing circle within a rhombus.
• There is no circumscribing circle around a rhombus.
• The diagonals of a rhombus lead to the formation of four right-angled triangles.
• These triangles are congruent to each other.
• Opposite angles of a rhombus are equal.
• When you connect the midpoint of the sides of a rhombus, a rectangle is formed.
• When the midpoints of half the diagonal are connected, another rhombus is formed.

**Diagonal of a Rhombus

A rhombus has four edges joined by vertices. On connecting the opposite vertices of a rhombus, additional edges are formed, which result in the formation of diagonals of a rhombus. Therefore, a rhombus can have two diagonals each of which intersects at an angle of 90°.

**Properties of diagonal of a rhombus

The diagonals of a rhombus have the following properties:

• The diagonals bisect each other at right angles.
• The diagonals of a rhombus divide into four congruent right-angled triangles.
• The diagonals of a rhombus may or may not be equal in length.

**Computation of diagonal of rhombus

The length of the diagonals of the rhombus can be calculated by using the following methods:

**By Pythagoras Theorem

Let us assume d1 to be the diagonal of the rhombus.

Since, we know, all adjacent sides in a rhombus subtend an angle of 90 degrees.

Therefore,

In the triangle, BCD we have,

BC2 + CD2 = BD2

Now, we have,

In the case of a square rhombus with all sides equal,

**Square Diagonal: a√2

where a is the length of the side of the square

In the case of a rectangle rhombus, we have,

**Rectangle Diagonal: √[l 2 + b 2 ]

where,

• l is the length of the rectangle.
• b is the breadth of the rectangle.

**By using the area of rhombus

Let us consider, O to be the point of intersection of two diagonals, namely d1 and d2.

Now,

The area of the rhombus is equivalent to,

A = 4 × area of ∆AOB

= 4 × (½) × AO × OB sq. units

= 4 × (½) × (½) d1 × (½) d2 sq. units

= 4 × (1/8) d1 × d2 square units

= ½ × d1 × d2

Therefore, **Area of a Rhombus = A = ½ × d 1 × d 2

**Area of rhombus using diagonals

Consider a rhombus ABCD, having two diagonals, i.e. AC & BD.

• **Step 1: Compute the length of the line segment AC, by joining the points A and C. Let this be diagonal 1, i.e. **d 1.

The diagonals of a rhombus are perpendicular to each other subtending right triangles upon intersection with each other at the centre of the rhombus.

• **Step 2: Similarly, compute the length of diagonal 2, i.e. **d 2 which is the distance between points B and D.
• **Step 3: Multiply both the calculated diagonals, d1, and d2.
• **Step 4: The result is obtained by dividing the product by 2.

The resultant will give the area of a rhombus ABCD.

Sample Questions

**Question 1. One of the sides of a rhombus is equivalent to 5 cm. One of the diagonals of the rhombus is 8 cm, compute the length of the other diagonal.

**Solution:

Let us consider, ABCD to be a rhombus, where AC and BD are the diagonals.

We have,

Side of the rhombus is 5 cm

BD = 8 cm

Since, we know that the diagonals of rhombus perpendicularly bisect each other.

∴ BO = 4cm

By Pythagoras theorem, we have,

In right angled △AOB,

⇒ (AB)2 = (AO)2 + (BO)2

⇒ (5)2 = (AO)2 + (4)2

⇒ 25 = (AO)2 + 16

⇒ (AO)2 = 9

∴ AO = 3cm

⇒ AC = 2 × 3 = 6 cm

∴ The length of other diagonal of the rhombus is equivalent to 6 cm.

**Question 2. Calculate the area of a rhombus with diagonals equivalent to 6 cm and 8 cm respectively.

**Solution:

We know,

Diagonal 1, d1 = 6 cm

Diagonal 2, d2 = 8 cm

Area of a rhombus, A = (d1 × d2) / 2

Substituting the values,

= (6 × 8) / 2

= 48 / 2

= 24 cm2

Hence, the area of the rhombus is 24 cm2.

**Question 3. A rectangular park has 10m length and breadth is 8m. Compute the diagonal of park.

**Solution:

We have,

Length = 100m
Breadth = 8 m

Computing diagonals, we obtain,

Rectangle Diagonal = √[l2 + b2]

= √[102 + 82 ]

= √[164]

= 12.80 m

**Question 4. A square rhombus has a side of 5 cm. Compute the length of diagonal.

**Solution:

We have,

Side of square, a = 5 units

Computing diagonals, we obtain,

Square Diagonal = a√2

= 5√2

= 7.07 cm

**Question 5. The area of rhombus is 315 cm² and its perimeter is 180 cm. Find the altitude of the rhombus.

**Solution:

We have,

Perimeter of rhombus = 180 cm

Calculating for the side of rhombus,

Side of rhombus,b = P/4 = 180/4 = 45 cm

Now,

Area of rhombus = b × h

Substituting the values,

⇒ 315 = 45 × h

⇒ h = 315/45

⇒ h =7 cm

Therefore, altitude of the rhombus is 7 cm.