The length of the diagonal of a square is 8 cm. Find its area (in c m 2 )

Options

A. 25
B. 64
C. 32
D. 16

Answer

To find the area of a square given the length of its diagonal, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the sides of the square.
The diagonal of a square can be related to its sides using the Pythagorean theorem. If the side length of the square is \( a \), then the diagonal \( d \) can be expressed as:
\[ d = a\sqrt{2} \]

### Step 2: Substitute the given diagonal length into the equation.
We are given that the diagonal \( d \) is 8 cm. Therefore, we can set up the equation:
\[ 8 = a\sqrt{2} \]

### Step 3: Solve for the side length \( a \).
To find \( a \), we can rearrange the equation:
\[ a = \frac{8}{\sqrt{2}} \]
To simplify this, we can multiply the numerator and denominator by \( \sqrt{2} \):
\[ a = \frac{8\sqrt{2}}{2} = 4\sqrt{2} \, \text{cm} \]

### Step 4: Calculate the area of the square.
The area \( A \) of a square is given by the formula:
\[ A = a^2 \]
Substituting \( a = 4\sqrt{2} \):
\[ A = (4\sqrt{2})^2 = 16 \times 2 = 32 \, \text{cm}^2 \]

### Step 5: State the final answer.
Thus, the area of the square is:
\[ \text{Area} = 32 \, \text{cm}^2 \]

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