Hard SAT Math Riddle: Find the Perimeter of the Shaded Area

This math riddle challenges you to find the perimeter of a shaded area. It’s similar to a difficult SAT question that many students struggle with. The riddle focuses on problem-solving and geometric reasoning, making it suitable for high school students preparing for standardized tests.

The Riddle

A square with sides of length 10 units has a circle inscribed within it. The circle touches all four sides of the square. What is the perimeter of the shaded area, which is the area between the circle and the square?

Solution

Here’s how to solve the riddle:

1. Find the diameter of the circle: Since the circle touches all sides of the square, its diameter is equal to the side length of the square, which is 10 units.
2. Find the radius of the circle: The radius is half the diameter, so it’s 10 units / 2 = 5 units.
3. Find the circumference of the circle: The circumference is calculated using the formula C = 2πr, where r is the radius. So, the circumference is 2π * 5 units = 10π units.
4. Find the perimeter of the square: The perimeter of a square is calculated using the formula P = 4s, where s is the side length. So, the perimeter is 4 * 10 units = 40 units.
5. Find the perimeter of the shaded area: The perimeter of the shaded area is the difference between the perimeter of the square and the circumference of the circle. So, the perimeter is 40 units – 10π units ≈ 9.73 units.

Key Concepts

• Inscribed Circle: A circle inscribed within a square touches all four sides of the square.
• Circumference: The distance around a circle.
• Perimeter: The total distance around a shape.

Why This Riddle Is Difficult

This riddle is challenging because it requires you to combine different geometric concepts. You need to understand the relationship between a square and an inscribed circle, know how to calculate circumference and perimeter, and be able to apply those concepts to find the perimeter of a complex shape.

Practice Makes Perfect

This riddle is a great way to practice your problem-solving and geometric reasoning skills. If you’re preparing for the SAT or other standardized tests, make sure to practice similar problems. The more you practice, the better you’ll become at solving these types of questions.