Rhombus - Definition, Angles, Properties, Formulas, Examples (2024)

What is Rhombus?

Rhombus is a quadrilateral with all equal sides. Since opposite sides of a parallelogram are equal so, rhombus is a special type of a parallelogram whose all sides are equal.

How is a Rhombus Different from a Square?

The difference between asquareand a rhombus is that allanglesof a square are right angles, but the angles of a rhombus need not be right angles.

So, a rhombus with right angles becomes a square.

We can say, “Every square is a rhombus but all rhombus are not squares.”

Real-life Examples

Rhombus can be found in a variety of things around us, such as finger rings, rhombus-shaped earring, the structure of a window glass pane, etc.

Properties of a Rhombus

Some of the properties of a rhombus are stated below.

• All sides of a rhombus are equal. Here, AB = BC = CD = DA.
• Diagonalsbisect each other at 90°. Here, diagonals AC and BD bisect each other at 90°.
• Opposite sides are parallel in a rhombus. Here, AB ∥ CD and AD ∥ BC.
• Opposite angles are equal in a rhombus. ∠A = ∠C and ∠B = ∠D.
• Adjacent angles add up to 180°.

∠A + ∠B = 180°

∠B + ∠C = 180°

∠C + ∠D = 180°

∠A + ∠D = 180°

• All the interior angles of a rhombus add up to 360°.
• Adjacent angles of a rhombus add up to 180°.
• The diagonals of a rhombus are perpendicular to each other. Here, AC ⟂ BD.
• The diagonals of a rhombus bisect each other. Here, DI = BI and AI = CI.
• A rhombus has rotational symmetry of 180 degrees (order 2). That is, a rhombus retains its original orientation when rotated by an angle 180 degrees.
• The diagonals of a rhombus are the only 2 lines of symmetry that a rhombus has. These divide the rhombus into 2 identical halves.

Area of a Rhombus

The area of a rhombus is the region enclosed by the 4 sides of a rhombus.

There are two ways to find the area of a rhombus.

1. Area of a Rhombus When its Base and Altitude are Known

Area of rhombus is calculated by finding the product of its base and corresponding altitude (height).

So, Area of rhombus = base × height = (b × h) square units.

2. Area of a Rhombus When its Diagonals are Known

When length of the diagonals of a rhombus are known, then its area is given by half of their product.

So, Area of rhombus = $\frac{(d1\times d2)}{2}$ square units; where d1 and d2 are the diagonals of a rhombus.

Perimeter of Rhombus

The perimeter of a rhombus is the total length of its boundaries. As all the four sides of a rhombus are equal, its perimeter is calculated by multiplying the length of its side by 4.

That is, Perimeter of a rhombus = 4 × a units; where ‘a’ is the length of the side of the rhombus.

Solved Examples on Rhombus

Example 1: The length of two diagonals of rhombus are 18 cm and 12 cm. Find the area of rhombus.

Solution:

Diagonal (d1) = 18 cm

Diagonal (d2) = 12 cm

Area of rhombus = $\frac{(d1\times d2)}{2}$ = $\frac{(18\times 12)}{2}$ sq.cm = 108 sq.cm

Example 2: Find the perimeter of the rhombus with its side measuring 15 cm.

Solution:

Length of side of rhombus (a) = 15 cm

Perimeter of rhombus = 4 × a = 4 × 15 cm = 60 cm

Example 3: The area of a rhombus is 56 sq. cm. If the length of one of its diagonals is 14 cm, find the length of the other diagonal.

Solution:

Area of rhombus = 56 sq.cm

d1 = 14 cm

We know, area of rhombus = $\frac{(d1+d2)}{2}$

⇒ 56 = $\frac{(14\times d2)}{2}$

⇒ 56 = 7 × d2

⇒ d2 = 56 ÷ 7

⇒ d2 = 8 cm

So, the second diagonal of the given rhombus measures 8 cm.

Example 4: In rhombus, ABCD, if ∠A = 60°, find the measure of all other angles.

Solution:

∠A + ∠B = 180° (Adjacent angles adds up to 180°)

60° + ∠B = 180° (Given, ∠A = 60°)

∠B = 180° – 60°

∠B = 120°

∠C = ∠A = 60° (Opposite angles are equal in a rhombus)

∠D = ∠B = 120° (Opposite angles are equal in a rhombus)

Practice Problems on Rhombus

Frequently Asked Questions on Rhombus