C = 2\pi (4\sqrt{2}) = 8\sqrt{2}\pi.

February 22, 2026

$C = 2\pi (4\sqrt{2}) = 8\sqrt{2}\pi.$

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Solution: The diagonal of the square is the diameter of the circle. Using the Pythagorean theorem, the diagonal $ d $ of a square with side length $ 8 $ is $ d = 8\sqrt{2} $. Thus, the radius $ r $ of the circle is half the diagonal:
r = rac{8\sqrt{2}}{2} = 4\sqrt{2}.
The circumference $ C $ of the circle is given by $ C = 2\pi r $. Substituting the value of $ r $:
Thus, the circumference of the circle is $ oxed{8\sqrt{2}\pi} $.
Question: A science fair judge is evaluating a student's model of a planet, shaped like a sphere with radius $ 2r $, and a supporting half-sphere with radius $ r $. What is the ratio of the volume of the planet to the volume of the half-sphere?
Solution: The volume of a sphere with radius $ 2r $ is:

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