# UVa 10772

## Summary

A regular polygon with $n$ vertices ($n$ is an even number) is inscribed in a circle of radius $r$ . Then every two vertices of the polygon are joined with straight lines, and the resulting regions are colored in $n/2$ different colors as shown in figures.

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Determine the area of the $k$ -th region. Regions are numbered from 1, starting from the innermost region.

## Explanation

If you study the figures carefully, you'll see that the boundary of every regions (except the outermost one) is a regular polygon with $n$ vertices, and its vertices are intersections of chords of the polygon.

More formally, let $V_{i}$ be the i-th vertex of the polygon: $V_{i}=(r\cos {\frac {2\pi i}{n}},r\sin {\frac {2\pi i}{n}})$ . And let $1\leq k<{\frac {n}{2}}$ be the number of region. Then the $j$ -th vertex of the boundary of $k$ -th region is the intersection of line segments $V_{j}V_{j+(n/2)-k}$ and $V_{j+1}V_{j+1+(n/2)-k}$ .

The area of $k$ -th region is the difference between inner areas of boundaries of $k$ -th and $k-1$ -st regions.

Area of regular polygon can be computed, if you know coordinates of its center O and one of vertices V. It equals ${\frac {n}{2}}(OV)^{2}\sin {\frac {2\pi }{n}}$ ## Input

14
10 8 1
10 8 2
10 8 3
10 8 4
50 12 1
50 12 2
50 12 6
100 40 1
100 40 2
100 40 3
100 40 10
100 40 15
100 40 19
100 40 20


## Output

48.5281
117.1573
117.1573
31.3166
538.4758
1471.1432
353.9816
193.7897
576.5974
945.2074
2462.3319
1878.1616
576.5974
129.0335