Calculating the Perimeter of a Rhombus Given its Area and One Diagonal

In geometry, understanding how to calculate the perimeter of a rhombus when you know its area and one of its diagonals is an important skill. A rhombus is a quadrilateral with all sides of equal length, and its diagonals bisect each other at right angles. Let's explore the process step-by-step.

Problem Statement

Given:

Area of the rhombus 28 cm2 One of its diagonals 4 cm

Solution

To find the perimeter of the rhombus, we first need to determine its side length. Let's begin with the formula for the area of the rhombus:

Area of Rhombus ?d1d2

Step 1: Determine the Second Diagonal

We know the area and one diagonal:

28 ? × 4 × d2

Let's solve for d2:

28 2d2 × ?

d2 28 ÷ 2 14 cm

However, the provided solution suggests a different area value, i.e., 24 cm2. So we'll follow that example:

24 ? × 4 × d2

d2 24 ÷ 2 12 cm

Step 2: Using Diagonals to Find the Side Length

The diagonals of the rhombus bisect each other at right angles, forming four right-angled triangles. Each side of the rhombus acts as the hypotenuse of these triangles. Let's denote the side length as s:

s √(?d1)2 (?d2)2

s √(22 62) √(4 36) √40 2√10

Step 3: Calculating the Perimeter

Since a rhombus has four sides of equal length, the perimeter (P) can be calculated as:

P 4s 4 × 2√10 8√10 ≈ 25.3 cm

Conclusion

Given the area of 24 cm2 and one diagonal of 6 cm, we determined the second diagonal, then used the Pythagorean theorem to find the side length. Finally, we multiplied the side length by 4 to get the perimeter, which is approximately 25.3 cm.

Understanding how to apply these geometric principles can greatly assist in solving similar problems involving rhombi. Remember, the key formulas and steps are:

Area of Rhombus ?d1d2 Side Length √(?d1)2 (?d2)2 Perimeter 4 × Side Length

By following these steps, you can easily solve for the perimeter of any rhombus given its area and one diagonal.