The diagonal of a square is 12 cm long, find its area and length of one side.

Answer

To find the area and the length of one side of a square given that its diagonal is 12 cm long, we can follow these steps:

### Step 1: Use the formula for the area of a square in terms of its diagonal.
The formula for the area \( A \) of a square in terms of its diagonal \( d \) is given by:
\[ A = \frac{1}{2} \times d^2 \]

### Step 2: Substitute the value of the diagonal into the formula.
Given that the diagonal \( d = 12 \) cm, we substitute this value into the formula:
\[ A = \frac{1}{2} \times (12)^2 \]

### Step 3: Calculate \( (12)^2 \).
Calculating \( (12)^2 \):
\[ (12)^2 = 144 \]

### Step 4: Calculate the area.
Now substitute \( 144 \) back into the area formula:
\[ A = \frac{1}{2} \times 144 = 72 \text{ cm}^2 \]

### Step 5: Find the length of one side of the square.
The area of a square can also be expressed in terms of its side length \( s \):
\[ A = s^2 \]
We already found that \( A = 72 \text{ cm}^2 \), so we set up the equation:
\[ s^2 = 72 \]

### Step 6: Solve for \( s \).
To find \( s \), take the square root of both sides:
\[ s = \sqrt{72} \]

### Step 7: Simplify \( \sqrt{72} \).
We can simplify \( \sqrt{72} \) as follows:
\[ \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \text{ cm} \]

### Final Result:
- The area of the square is \( 72 \text{ cm}^2 \).
- The length of one side of the square is \( 6\sqrt{2} \text{ cm} \).

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