Finding the Perimeter of a Rhombus Given Its Area and One Diagonal

A rhombus is a type of quadrilateral with all sides of equal length. It is interesting not only for its symmetrical properties but also for the formulas we can use to calculate various measurements, such as the perimeter, given certain pieces of information. In this article, we will explore how to determine the perimeter of a rhombus when the area and one of its diagonals are known. We will follow the steps to find the missing diagonal, calculate the side length of the rhombus, and ultimately find its perimeter.

Understanding the Rhombus and Its Properties

A rhombus is defined by its properties, including:

All four sides are of equal length. The diagonals bisect each other at right angles.

To find the perimeter of a rhombus, we need to first determine the length of one of its sides. This involves using the given area and one of its diagonals.

Given Information and Initial Setup

In this problem, we are given that the area of a rhombus is 36 cm2 and one of its diagonals is 18 cm. To proceed, we can use the formula for the area of a rhombus:

Area of a Rhombus: A (d1 × d2) / 2

Calculating the Other Diagonal

Using the given area and one diagonal, we can set up the equation:

A (18 × d2) / 2 36 cm2

Now, let's solve for the other diagonal (d2):

18 × d2 72

d2 72 / 18 4 cm

Dividing the Diagonals into Half

Since the diagonals of a rhombus bisect each other at right angles, we can divide each diagonal into two equal parts:

d1/2 18/2 9 cm d2/2 4/2 2 cm

Calculating the Side Length of the Rhombus

Each of the four right-angled triangles formed by the diagonals has legs of 9 cm and 2 cm. To find the side length of the rhombus, we use the Pythagorean theorem:

s2 92 22 81 4 85

s √85 cm

Thus, the length of each side of the rhombus is approximately 9.22 cm.

Calculating the Perimeter of the Rhombus

The perimeter of a rhombus is given by the formula:

P 4s

Substituting the side length we found:

P 4 × √85 ≈ 4 × 9.22 ≈ 36.88 cm

Final Answer and Conclusion

Therefore, the perimeter of the rhombus is approximately 36.88 cm. This calculation is a practical application of the properties of a rhombus, particularly using its diagonals and area to determine the side lengths and, ultimately, the perimeter.

Understanding the relationships between the diagonals and the area of a rhombus can be a valuable tool in solving geometric problems. Whether you are working on a geometry project or preparing for a math exam, this method can help you tackle similar problems with ease.