The area of a rhombus is 216 sq. cm. If its one diagonal is 24 cm, find :
perimeter of the rhombus.

Answer

To find the perimeter of the rhombus given its area and one diagonal, we can follow these steps:

### Step 1: Understand the given information
We know:
- Area of the rhombus = 216 sq. cm
- One diagonal (AC) = 24 cm

### Step 2: Use the formula for the area of a rhombus
The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

### Step 3: Substitute the known values into the formula
Here, we can substitute the known values into the area formula:
\[ 216 = \frac{1}{2} \times 24 \times d_2 \]

### Step 4: Solve for the unknown diagonal (BD)
First, simplify the equation:
\[ 216 = 12 \times d_2 \]
Now, divide both sides by 12 to find \( d_2 \):
\[ d_2 = \frac{216}{12} = 18 \text{ cm} \]

### Step 5: Find the lengths of the halves of the diagonals
Since the diagonals bisect each other at right angles in a rhombus, we can find the lengths of the halves:
- Half of diagonal AC (OA) = \( \frac{24}{2} = 12 \text{ cm} \)
- Half of diagonal BD (OB) = \( \frac{18}{2} = 9 \text{ cm} \)

### Step 6: Use the Pythagorean theorem to find the side length of the rhombus
In triangle OAB, we can apply the Pythagorean theorem:
\[ AB^2 = OA^2 + OB^2 \]
Substituting the values:
\[ AB^2 = 12^2 + 9^2 = 144 + 81 = 225 \]
Now, take the square root to find AB:
\[ AB = \sqrt{225} = 15 \text{ cm} \]

### Step 7: Calculate the perimeter of the rhombus
Since all sides of a rhombus are equal, the perimeter \( P \) can be calculated as:
\[ P = 4 \times AB = 4 \times 15 = 60 \text{ cm} \]

### Final Answer
The perimeter of the rhombus is **60 cm**.

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