Finding Square Diagonal Length from Area

The problem asks us to calculate the length of the diagonal of a square given its area. We are given the area of the square as \(32\ cm^2\).

Square Properties and Formulas

For a square with side length \(s\) and diagonal length \(d\), the following relationships hold:

• Area (\(A\)) = \(s^2\)
• Diagonal (\(d\)) = \(s\sqrt{2}\)

We can also relate the area directly to the diagonal. Since \(s = \frac{d}{\sqrt{2}}\), the area can be expressed as:

\(A = s^2 = \left(\frac{d}{\sqrt{2}}\right)^2 = \frac{d^2}{2}\)

Rearranging this formula to solve for the diagonal \(d\), we get:

\(d^2 = 2A\)

\(d = \sqrt{2A}\)

Step-by-Step Calculation

1. Identify the given Area: The area (\(A\)) of the square is \(32\ cm^2\).
2. Use the Area-Diagonal Formula: Apply the formula \(d = \sqrt{2A}\) to find the diagonal length.
3. Substitute and Calculate: Substitute the given area into the formula:

   \(d = \sqrt{2 \times 32\ cm^2}\)

   \(d = \sqrt{64\ cm^2}\)

   \(d = 8\ cm\)

Conclusion

The length of the diagonal of the square is \(8\ cm\). This matches option A.