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# Area of Rhombus – Explanation & Examples

We saw in the Polygon article that the **rhombus is a quadrilateral with four parallel sides of equal lengths**. The opposite angles of a rhombus are also equal.

Similarly, **the diagonals of a rhombus intersect at right angles, and their lengths are always equal**. A square is a type of rhombus whose 4 angles are all right angles. Sometimes, a rhombus is referred to as a rhomb, diamond, or lozenge.

In this article, you will learn how to calculate a rhombus area using the three areas of rhombus formulas.

## How to Calculate the Area of a Rhombus?

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

*There are three ways to find the area of a rhombus.*

**One way** is by the use of the altitude and side of a rhombus. **The second method** entails using the side and angle, and **the last method entails using** the diagonals.

These formulas for calculating the area of a rhombus are collectively known as rhombus area formulas. Let’s take a look.

## Rhombus Area Formula

We can find the area of the rhombus in multiple ways. We will saw each of them one by one below.

### Area of Rhombus using Altitude and Base

When the altitude or height and the length of the sides of a rhombus are known, the area is given by the formula;

Area of rhombus = base × height

**A = b × h**

Let us see understand this through an example:

*Example 1*

Find the area of a rhombus whose side is 30 cm and height is 15 cm.

__Solution__

A = b × h

= (30 x 15) cm^{2}

= 450 cm^{2}

Therefore, the area of the rhombus is 450 cm^{2}.

*Example 2*

Calculate the area of the rhombus shown below.

__Solution__

A = b × h

= (18 x 24) mm^{2}

*Example 3*

If the height and area of a rhombus are 8 cm and 72 cm^{2,} respectively, find the rhombus’s dimensions.

__Solution__

A = b × h

72 cm^{2} = 8 cm x b

Divide both sides by 8.

72 cm^{2}/8 cm = b

b = 9 cm.

Therefore, the dimensions of the rhombus are 9 cm by 9 cm.

*Example 4*

The base of a rhombus is 3 times plus 1 more than the height. If the area of the rhombus is 10 m^{2}, find the base and height of the rhombus.

__Solution__

Let the height of rhombus = x

and base = 3x + 1

A = b × h

10 m^{2} = x (3x + 1)

10 = 3x^{2} + x

3x^{2} + x – 10 = 0

Solve the quadratic equation.

⟹ 3x^{2} + x – 10 = 3x^{2 }+ 6x – 5x – 10

⟹ 3x (x + 2) – 5(x + 2)

⟹ (3x – 5) (x + 2) = 0

⟹ 3x – 5 = 0

⟹ x = 5/3

⟹ x + 2= 0

x = -2

Now substitute the value of x.

Height = x = 5/3 m

Base = 3x + 1 = 3(5/3) + 1 = 6 m

So, the base of the rhombus is 6 m, and the height is 5/3 m.

### Area of Rhombus Using Diagonals

Given the lengths of the diagonals, the area of a rhombus is equal to half the diagonals’ product.

**A = ½ × d _{1} × d_{2}**

Where d_{1} and d_{2 }are the diagonals of a rhombus.

*Example 5*

The two diagonals of a rhombus are 12 cm and 8 cm. Calculate the rhombus area.

Solution:

Let d_{1} = 12 cm and d_{2} = 8 cm.

A = ½ × d_{1} × d_{2}

= (½ × 12 × 8) cm^{2}.

= 48 cm^{2}.

*Example 6*

Calculate the side lengths if its area is 24 cm^{2}, diagonal is 8 cm, and height 3 cm.

__Solution__

Let d_{1} = 8 cm.

d_{2} =?

A = ½ × d_{1} × d_{2}

24 cm^{2} = ½ × 8 × d_{2}

24 cm^{2} = 4d_{2}

Divide both sides by 4 to get,

6 = d_{2}

Therefore, the other diagonal is 6 cm.

Now, calculate the side lengths of the rhombus.

A = b × h

24 cm^{2} = 3 cm x b

Divide both sides by 3.

8 cm = b.

Therefore, the side lengths of the rhombus are 8 cm.

*Example 7*

Find the diagonals of the rhombus shown below if its area is 3,458 cm^{2}.

__Solution__

A = ½ × d_{1} × d_{2}

3,458 cm^{2} = ½ * 6x * 8x

3,458 cm^{2} = 24x^{2}

Divide both sides by 24.

3,458/24 = x^{2}

144 = x^{2}

Find the square root of both sides.

x = -12 or 12.

Length cannot be a negative number; therefore, substitute only x =12 in the diagonals’ equations.

6x = 6 * 12 = 72 cm

8x = 8 * 12 = 96 cm

Hence, the lengths of the diagonals are 72 cm and 96 cm.

*Example 8*

Suppose the rate of polishing a floor is $ 4 per square meter. Find the cost of polishing a rhombus-shaped floor, and each of its diagonals is 20 m and 12 m.

__Solution__

To find the cost of polishing the floor, multiply the polishing rate by the area of rhombus shaped floor.

A = ½ × 20 m × 12 m

= 120 m^{2}

Cost of painting = 120 m^{2} x $ 4 per m.

= $480

### Area of Rhombus using the Length of the Sides and an included Angle.

The area of a rhombus is equal to the product side length squared and the sine of the angle between the two sides.

**Area of rhombus = b ^{2} × Sine (A)**

Where A = angle formed between two sides of a rhombus.

*Example 9*

Find the area of a rhombus whose sides are 8 cm, and the angle between the two sides is 60 degrees.

__Solution__

A = b^{2} × Sine (A)

= 8^{2} x sine (60)

= 55.43 cm^{2}.

*Practice Questions*

*Practice Questions*

- Find the length of a diagonal of a rhombus if the other diagonal is 5 units long, and the area of a rhombus is 30 square units.
- A kite has a shorter diagonal of length 16 units, a shorter side of length 10 units, and a longer side of length 17 cm. What is the length of the other diagonal?
- What area of a rhombus whose side lengths are 18 cm each and one diagonal is 20 cm?