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