# 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) cm2

= 450 cm2

Therefore, the area of the rhombus is 450 cm2.

Example 2

Calculate the area of the rhombus shown below.

Solution

A = b × h

= (18 x 24) mm2

Example 3

If the height and area of a rhombus are 8 cm and 72 cm2, respectively, find the rhombus’s dimensions.

Solution

A = b × h

72 cm2 = 8 cm x b

Divide both sides by 8.

72 cm2/8 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 m2, find the base and height of the rhombus.

Solution

Let the height of rhombus = x

and base = 3x + 1

A = b × h

10 m2 = x (3x + 1)

10 = 3x2 + x

3x2 + x – 10 = 0

Solve the quadratic equation.

⟹ 3x2 + x – 10 = 3x2 + 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 × d1 × d2

Where d1 and d2 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 d1 = 12 cm and d2 = 8 cm.

A = ½ × d1 × d2

= (½ × 12 × 8) cm2.

= 48 cm2.

Example 6

Calculate the side lengths if its area is 24 cm2, diagonal is 8 cm, and height 3 cm.

Solution

Let d1 = 8 cm.

d2 =?

A = ½ × d1 × d2

24 cm2 = ½ × 8 × d2

24 cm2 = 4d2

Divide both sides by 4 to get,

6 = d2

Therefore, the other diagonal is 6 cm.

Now, calculate the side lengths of the rhombus.

A = b × h

24 cm2 = 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 cm2.

Solution

A = ½ × d1 × d2

3,458 cm2 = ½ * 6x * 8x

3,458 cm2 = 24x2

Divide both sides by 24.

3,458/24 = x2

144 = x2

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 $4 per square meter. Find the cost of polishing a rhombus-shaped floor, and each of its diagonals is 20 m and 12 m. Solution To find the cost of polishing the floor, multiply the polishing rate by the area of rhombus shaped floor. A = ½ × 20 m × 12 m = 120 m2 Cost of painting = 120 m2 x$ 4 per m.

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