The Definition and Formula for Finding the Area of Rhombus

If you've ever seen a diamond-shaped kite flying in the sky on a windy day, then you're familiar with the shape of a rhombus. In Geometry, a rhombus is defined as a four-sided shape with four equal sides and opposite sides that are parallel to each other. This unique shape falls under the category of both a parallelogram and a quadrilateral, with its distinguishing feature being that all four sides are equal.

The area of a rhombus can be found using a specific formula. Let's consider a rhombus with diagonals of length d1 and d2.

Area of a Rhombus Formula

The formula for finding the area of a rhombus is:

Area = (d1 x d2) / 2

As a rhombus falls within the classification of parallelograms, the formula for finding the area of a parallelogram can also be used for a rhombus. Let's take a look at a parallelogram with a base of b and a height of h.

Area of Parallelogram Formula

The formula for finding the area of a parallelogram is:

Area = b x h

Where b is the base and h is the height.

In the case of a rhombus, any side can be considered the base since all sides are equal. This also means that the height or altitude will be the same regardless of which side is chosen as the base. However, for a parallelogram, the convention is to use the longer sides as the base. The example of a parallelogram above illustrates that its sides are not equal, making it distinct from a rhombus.

It's worth noting that when calculating the area, we use square units. For instance, if you're using SI units, then the area unit would be square meters (m²).

Examples of Finding the Area of a Rhombus

Let's explore a few problems related to determining the area of a rhombus.

Example 1:

Given a rhombus WXYZ with an area of 138 in², find the length in inches of the diagonal XZ.

Solution:
We know that for a rhombus, the diagonal and the sides are related by the following formula:

d² = a² + b²

Where d is the diagonal, and a and b are the sides.

Therefore:

138² = a² + b²

Since all sides are equal in a rhombus, we can substitute a with the length of either side.

138² = 4a²

Solving for a, we get a = 11.75.

Since the diagonal XZ is equal to the sides, the length of the diagonal is also 11.75 inches.

Example 2:

A rhombus has diagonals of lengths 6 m and 7 m. What is the area of the rhombus?

Solution:
Using the same formula, we can substitute the values of d1 and d2 to get the formula:

Area = (6 x 7) / 2

Area = 21 m²

Example 3:

Given a rhombus with an area of 100 sq. units and an altitude of 8 units, what is the length of the sides of the rhombus?

Solution:
Since a rhombus is also a parallelogram, we can use the formula for the area of a parallelogram.

Area = base x height

100 = base x 8
=> base = 12.5 units

This means that the length of the sides of the rhombus is 12.5 units.

Special Case: The Area of a Square

A square is another type of rhombus, with all four sides equal. Additionally, all four angles of a square are right angles, making it a special case of a rhombus and a rectangle. As with a rhombus, the formula for the area of a parallelogram can also be applied to squares as they are also parallelograms.

Area = side x side

Example:

Given a square with an area of 64 square units, find the length of the sides and diagonals of the square.

Solution:
Area = side²
64 = side²
=> side = 8

This means that the length of the sides and diagonals of the square is 8 units.

Finding the Area of a Rhombus: Formulas and Properties Explained

In geometry, a rhombus is a four-sided figure with equal sides, making it a special type of quadrilateral. One common task in working with shapes is determining their area, and the same goes for a rhombus. To do so, we use the formula Area = base × height, where the base is represented by b and the height by h. This formula is also applicable to a parallelogram, so it can be used to find the area of a rhombus. However, there is another commonly used formula specifically for calculating the area of a rhombus.

If we have a rhombus with diagonals of length d1 and d2, we can use the formula Area = 0.5 × d1 × d2 to determine its area. This formula is derived from the fact that a rhombus can be divided into four triangles, each with a base of one diagonal and a height of half the other diagonal. By multiplying 0.5 by both diagonals, we can find the total area of the rhombus.

But what if we only know the length of one side and one diagonal of a rhombus? In this case, we can still find the area by using the Pythagorean theorem to calculate the length of the other diagonal. Then, we can use one of the aforementioned formulas, depending on the given information, to determine the area of the rhombus.

Key Takeaways About the Properties of a Rhombus

• The diagonals of a rhombus are always perpendicular to each other.
• If the diagonals of a parallelogram are perpendicular, it indicates that the parallelogram is actually a rhombus.
• Each diagonal of a rhombus bisects a pair of opposite angles. This means that the diagonals divide the rhombus into four equal triangles.

In summary, a rhombus is a specific type of quadrilateral with four equal sides. To calculate its area, we can use either the formula Area = 0.5 × d1 × d2 or the formula for a parallelogram, Area = base × height. It's also important to remember the properties of a rhombus, such as the perpendicular diagonals and the fact that each diagonal bisects its opposite angles. With this knowledge, finding the area of a rhombus should be a straightforward task in any given situation.