SAT Math Multiple Choice Practice Question 416: Answer and Explanation
20. A square is inscribed in a circle with radius r. What is the probability that a randomly selected point within the circle will NOT be within the square?
Correct Answer: B
B Probability is the chance of something happening. In this case, to find the probability, find the area that is in the circle but not the square, divided by the area of the circle (which represents all possibilities). Plug in for the radius of the circle. Let's say r = 5. So, the area of the circle is π × 52 = 25π. The area that is in the circle but not the square is the area of the circle minus the area of the square. Find the area of the square. The diagonal of the square is equal to the diameter of the circle: 2 × 5 = 10. The diameter creates a 45°-45°-90° triangle from the square. So, the side of the square is . That means the area of the square is . Therefore, the area within the circle but not the square is 25π – 50. That means the probability is . This is the target. Only B matches. After you've plugged in and realized that the probability is , you can also solve algebraically:
Another, even more complicated, approach would be to call the area of the circle πr2, the diameter 2r, and the area of the square . The probability would be . This is simplified: . It's much more confusing when you don't plug in numbers!