The area of the square that can be inscribed in a circle of radius 8 cm is
\(256\,\text{cm}^2\)
\(128\,\text{cm}^2\)
\(64\sqrt{2}\,\text{cm}^2\)
\(64\,\text{cm}^2\)
Step 1: The circle has radius \(r = 8\,\text{cm}\). So, the diameter of the circle is:
\( \text{Diameter} = 2r = 2 \times 8 = 16\,\text{cm} \).
Step 2: A square inscribed in a circle means that the circle passes through all four corners of the square. In this case, the diagonal of the square is exactly equal to the diameter of the circle.
So, \( \text{Diagonal of square} = 16\,\text{cm} \).
Step 3: For a square, the relation between diagonal (d) and side (a) is:
\( d = a\sqrt{2} \).
Step 4: Substituting the diagonal value:
\( a\sqrt{2} = 16 \).
Step 5: Solve for side (a):
\( a = \dfrac{16}{\sqrt{2}} = 8\sqrt{2}\,\text{cm} \).
Step 6: The area of a square is given by:
\( \text{Area} = a^2 \).
Step 7: Substitute side length:
\( \text{Area} = (8\sqrt{2})^2 = 64 \times 2 = 128\,\text{cm}^2 \).
Final Answer: The area of the inscribed square is 128 cm². So, the correct option is (B).