Circles

In the triangle $KLM$, $k=10\mathrm{cm}$, $l=8\mathrm{cm}$, $m=12\mathrm{cm}$. Point $N$ is the foot of the altitude from the vertex $K$ (See the picture.) What is the radius of the circumcircle of the triangle $KLN$?

Consider a right-angled triangle $ABC$ with the right angle at the vertex $C$. Calculate the measure of the angle $CAB$, if the side $b=9\mathrm{cm}$ and the radius of the circumscribed circle $r=6\mathrm{cm}$. Round the result to one decimal place.

The curved surface of a cone has an area of $4.15{\mathrm{cm}}^2$. If we flatten it into a plain, we get a sector of a circle with the central angle of ${126}^{\circ }$. Calculate the volume of this cone. Round the result to two decimal places.

Find the length of the diagonal of a square that has the same area as the circle with a radius of $10\mathrm{cm}$.