Area of Circle Union

We want to calculate the area of the union of a set of circles. This is a problem that has some non-trivial algorithm for the exact computation. Instead, we would be satisfied by finding an approximation using a Montecarlo method. The ideas is as follows:

Calculate a bounding box around the circles.
Generate random points within the bounding box.
Count how many points are inside some circle.

0.4

0.1 $\Longrightarrow$

0.4

Exam score: 2.5 Automatic part: 100%

Input

The input contains a set of cases. Each case specifies the number of circles, $n\geq 0$ , and the number of random points generated for the Montecarlo approximation. After that, a list of $n$ circles is specified, each one with the coordinates of the center, $(x,y)$ , and the radius. The coordinates and the radius are real numbers.

Output

For every case print the estimated area as a real number in free format.

Observation

There is no need to compute the exact area. The output will be considered correct if it is a good approximation of the area.

Problem information

Author: Professors de Info-FME

Generation: 2026-01-25T21:46:48.192Z