Final answer:
The diagonal of the square is the same as the diameter of the circle, which is 36 cm, and using the Pythagorean theorem, the side of the square is found. The area of the square is then calculated to be approximately 648 cm².
Step-by-step explanation:
To determine the area of the largest square that can be cut out of a circle with a radius of 18 cm, we need to understand the relationship between the diameter of the circle and the side of the square. The diagonal of the square is equal to the diameter of the circle because the largest square that fits inside the circle will touch the circle at all four corners. Since the diameter of the circle is twice the radius, the diagonal of the square (d) is 36 cm (2 × 18 cm).
According to the Pythagorean theorem, for a square with side length 's' and diagonal 'd', we have d² = s² + s², simplified to d² = 2s². Therefore, s = d/√2 = 36 cm/√2 = 36 cm/1.414 ≈ 25.455 cm. The area of the square (A) is s², which is approximately 25.455 cm × 25.455 cm ≈ 648 cm².
Therefore, the area of the largest square that can be cut out of a circle of radius 18cm is approximately 648 cm², which corresponds to option (C).