The Queen's Terrace

The queen wants a terrace garden. The garden is one stone in the middle and several rings of stones around it.

• The middle stone is a circle of radius $1$.
• Each ring is built from $N$ circular stones of equal size laid out around the middle. Every ring holds the same number $N$ of stones, but the stone size changes from ring to ring.
• Two neighboring stones in the same ring touch each other.
• The stones of the first ring touch the middle stone.
• From the second ring on, each stone touches a stone of the ring just inside it. An outer ring is rotated by half a step against the ring inside it, so one stone of the outer ring touches two neighboring stones of the inner ring at the same time.

The queen has not fixed $N$ and $M$ yet, so she wants the size of the terrace worked out in advance for many combinations. The terrace is the shortest convex boundary that encloses every stone of the outermost ring. That boundary is made of the outer arc of each stone and the segments joining the arc ends of neighboring stones.

For a garden with $M$ rings, find the radius of a stone in the last ring and the perimeter of the terrace.

The first line holds the number of test cases $P$. ($1 \le P \le 1000$)

Each of the next $P$ lines holds the test case number $T$, the number of stones $N$ in one ring, and the number of rings $M$, separated by spaces. ($3 \le N \le 20$, $1 \le M \le 15$) $T$ is an integer.

For each test case print one line with the test case number $T$, the radius of a stone in the last ring, and the perimeter of the terrace, separated by single spaces.

Round both real numbers at the fourth decimal place and print three decimal places. Rounding is half up, and an integral value still shows all three decimals. The output must match the answer character for character.

The answer grows as $N$ gets smaller and $M$ gets larger. The radius reaches about $5.6 \times 10^{14}$ and the perimeter about $6.9 \times 10^{15}$, so double precision alone cannot pin down the third decimal. Compute with higher precision.