UVa 10772
10772  Rose Windows[edit]
Summary[edit]
A regular polygon with vertices ( is an even number) is inscribed in a circle of radius . Then every two vertices of the polygon are joined with straight lines, and the resulting regions are colored in different colors as shown in figures.
Error creating thumbnail: Unable to save thumbnail to destination 
Error creating thumbnail: Unable to save thumbnail to destination 

n=8  n=12 
Determine the area of the th region. Regions are numbered from 1, starting from the innermost region.
Explanation[edit]
If you study the figures carefully, you'll see that the boundary of every regions (except the outermost one) is a regular polygon with vertices, and its vertices are intersections of chords of the polygon.
More formally, let be the ith vertex of the polygon: . And let be the number of region. Then the th vertex of the boundary of th region is the intersection of line segments and .
The area of th region is the difference between inner areas of boundaries of th and st regions.
Area of regular polygon can be computed, if you know coordinates of its center O and one of vertices V. It equals
Input[edit]
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[edit]
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