Determine the type of body obtained by rotating a square around its diagonal.

## Answer

Two equal cones with a common base.

Find the area of the section of a sphere with radius 3 by a plane that is 2 units away from its center.

#
Let $O$ be the center of the given sphere, $O_1$ be the base of the perpendicular dropped from point $O$ to the plane of the section. Then $O_1$ is the center of the circle that is the section of the sphere. Let $M$ be an arbitrary point on the circumference of this circle, $r$ the radius of the circle, $S$ the area. Then $O_1M$ is the radius of the section. By the Pythagorean theorem from the right triangle $OO_1M$, we find that

$$
O_1M^2 = OM^2 - OO_1^2 = 9 - 4 = 5.
$$

Therefore,

$$
S = \pi r^2 = \pi \cdot O_1M^2 = 5\pi
$$

## Answer

$5\pi$.