Rose windows

Mr. Arnold Gerald Nostik is in charge of the design of the main rose
window of the new cathedral in his town. The rose window is circular, 2r
units wide. Since Mr. A. G. Nostik knows little about Virgins, Saints
and Angels, he is thinking about a geometric pattern. Let n be an even
integer number, at least 4. Mr. Nostik plans to pick n points, each at
distance r of the center of the window, these points being the vertices
of a regular polygon. (The next page shows an example with n = 8.) These
points are to be joined with straight lines, and the resulting regions
colored (with arbitrary colors) as shown in the example. Note that for
n = 8 there are four regions. We number these regions 1, 2, 3 and 4
starting to count from the center of the rose. In general, there are n/2
regions.

Write a program to help Mr. Nostik to know how much glass of every color
he needs in order to build a given rose window.

Input

Input begins with a number t. Follow t cases, each with a real number r,
an even integer number n, and an integer number k. Assume 1 ≤ r ≤ 100,
4 ≤ n ≤ 40, and 1 ≤ k ≤ n/2.

Output

For every case, print the area of the k-th region of a rose window with
n points and radius r, with four digits after the decimal point. The
input cases have no precision issues.

[image]

Problem information

Author: Salvador Roura

Generation: 2026-01-25T11:54:29.064Z

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