The side of a rhombus is 15cm. If its one diagonal is 18 cm. Find its area.

Answer

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

### Step 1: Understand the properties of the rhombus
A rhombus has all sides equal and its diagonals bisect each other at right angles. We are given:
- Side of the rhombus (s) = 15 cm
- One diagonal (d1) = 18 cm

### Step 2: Use the relationship between the diagonals and the sides
Let the other diagonal be \(d2\). In a rhombus, the relationship between the sides and the diagonals is given by:
\[ s^2 = \left(\frac{d1}{2}\right)^2 + \left(\frac{d2}{2}\right)^2 \]
Substituting the known values:
\[ 15^2 = \left(\frac{18}{2}\right)^2 + \left(\frac{d2}{2}\right)^2 \]
\[ 225 = 9^2 + \left(\frac{d2}{2}\right)^2 \]
\[ 225 = 81 + \left(\frac{d2}{2}\right)^2 \]

### Step 3: Solve for \(d2\)
Rearranging the equation gives:
\[ \left(\frac{d2}{2}\right)^2 = 225 - 81 \]
\[ \left(\frac{d2}{2}\right)^2 = 144 \]
Taking the square root of both sides:
\[ \frac{d2}{2} = 12 \]
Thus, \(d2 = 24\) cm.

### Step 4: Calculate the area of the rhombus
The area \(A\) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d1 \times d2 \]
Substituting the values of the diagonals:
\[ A = \frac{1}{2} \times 18 \times 24 \]
Calculating this gives:
\[ A = \frac{1}{2} \times 432 = 216 \text{ cm}^2 \]

### Final Answer
The area of the rhombus is \(216 \text{ cm}^2\).

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