Mystery of the Week: How big is the dark area?

A mathematics puzzle has sparked widespread interest online after a German publication presented it as part of its weekly riddle series. The question asks readers to determine the area of the shaded region within a rectangle, given specific distances between the centers of three circles arranged inside it. According to the solution published by Der Spiegel, the shaded area measures approximately 88.6 square units. The problem describes two larger circles placed side by side within a rectangle, with a smaller circle touching both of them. The distances between the centers of the three circles are given as 7, 9, and 12 units. Solving this requires calculating the radii of each circle based on these distances and using geometric principles to find the total area of the rectangle and subtracting the areas of the circles to arrive at the final answer. To begin, the distance between the centers of the two larger circles is 12 units. Since their centers are separated by this distance, and one circle's radius is 2 units greater than the other, the radii can be calculated. This leads to the determination that the larger circle has a radius of 7 units and the middle-sized circle has a radius of 5 units. The smaller circle, which touches both larger ones, has a radius of 2 units. Using the Pythagorean theorem, the height of the rectangle is determined by considering the vertical distance between the centers of the two larger circles and the small circle. This calculation involves finding the square root of 140, resulting in a value of approximately 11.83 units. Adding twice this value to the width of the rectangle, calculated as 14 units, gives the full dimensions of the rectangle. With the dimensions known, the area of the rectangle is computed as 14 multiplied by the sum of 12 and twice the square root of 35, yielding approximately 333.65 square units. From this, the combined area of all three circles is subtracted. The individual areas of the circles are calculated using the formula πr², leading to a total area of approximately 245.04 square units. Subtracting the circular areas from the rectangle’s area gives the final result for the shaded region, which is rounded to 88.6 square units. The solution provides step-by-step calculations, including detailed mathematical derivations, ensuring clarity for those attempting to solve the problem independently. This type of geometric puzzle is common in recreational mathematics and often appears in publications aimed at engaging readers with logical reasoning and spatial awareness. While the exact origin of the problem is unclear, similar puzzles have appeared in various media outlets over the years, suggesting a long-standing tradition of using such challenges to stimulate intellectual curiosity. The solution, while mathematically rigorous, relies on basic geometry and algebraic manipulation. Readers who attempted the problem independently were encouraged to verify their results against the provided steps, reinforcing the educational value of such exercises. The widespread sharing of the solution online indicates a growing appreciation for problems that blend visual logic with numerical precision.