The diagonal of the square is \(4\sqrt{2}\text{ cm}\). Therefore its side is

\(a=\dfrac{4\sqrt{2}}{\sqrt{2}}=4\text{ cm}\).

The axial section of a cylinder is a rectangle with sides \(h\) and \(2r\). According to the condition it is a square, so

\(h=2r=4\).

Hence \(r=2\text{ cm}\), \(h=4\text{ cm}\).

The total surface area of the cylinder is

\(S_{\text{tot}}=2\pi r^2+2\pi rh=2\pi r(r+h)\).

Substitute the data and \(\pi=3\):

\(S_{\text{tot}}=2\cdot 3\cdot 2\cdot (2+4)=72\text{ cm}^2\).

Answer: \(72\).

The base area of the cylinder is \(\pi r^2\). According to the condition,

\(\pi r^2=9\pi\).

So \(r^2=9\), hence \(r=3\text{ cm}\).

The axial section of the cylinder is a rectangle with sides \(2r\) and \(h\).

Therefore its area is

\(S=2r\cdot h=2\cdot 3\cdot 5=30\text{ cm}^2\).

Answer: \(30\).

If a sphere is inscribed in a cube, then its diameter equals the edge of the cube:

\(d=4\text{ cm}\).

Therefore the radius is

\(r=2\text{ cm}\).

The volume of a sphere is

\(V=\dfrac{4}{3}\pi r^3\).

Since \(\pi=3\), we get

\(V=\dfrac{4}{3}\cdot 3\cdot 2^3=4\cdot 8=32\text{ cm}^3\).

Answer: \(32\).

A plane section of a sphere is a circle. If the sphere radius is \(R=6\text{ cm}\), and the distance from the center to the plane is \(d=4\text{ cm}\), then the radius \(r\) of the section is found from the Pythagorean theorem:

\(r^2=R^2-d^2=6^2-4^2=36-16=20\).

Therefore the area of the section is

\(S=\pi r^2=\pi\cdot 20\).

For \(\pi=3\):

\(S=3\cdot 20=60\text{ cm}^2\).

Answer: \(60\).

If a sphere is inscribed in a cylinder, then the cylinder radius equals the sphere radius, and the cylinder height equals the sphere diameter.

So

\(r=1\text{ cm},\quad h=2\text{ cm}\).

The volume of the cylinder is

\(V=\pi r^2h\).

Substitute the values and \(\pi=3\):

\(V=3\cdot 1^2\cdot 2=6\text{ cm}^3\).

Answer: \(6\).

For a right cylinder, the generatrix equals the height, so

\(h=10\text{ cm}\).

A section by a plane parallel to the axis is a rectangle. One side equals the cylinder height, and the other side is a chord of the base.

The distance from the center of the base circle to the chord is \(12\text{ cm}\), and the radius is \(13\text{ cm}\).

Half of the chord is

\(x=\sqrt{13^2-12^2}=\sqrt{169-144}=5\text{ cm}\).

So the chord length is

\(2x=10\text{ cm}\).

The area of the section is

\(S=10\cdot 10=100\text{ cm}^2\).

Answer: \(100\).

A sphere passing through all vertices of a cube is circumscribed about the cube. The radius of such a sphere equals half the space diagonal of the cube.

The space diagonal of the cube is

\(d=a\sqrt{3}=2\sqrt{3}\cdot \sqrt{3}=6\text{ cm}\).

Hence the sphere radius is

\(R=\dfrac{d}{2}=3\text{ cm}\).

The volume of the sphere is

\(V=\dfrac{4}{3}\pi R^3\).

For \(\pi=3\):

\(V=\dfrac{4}{3}\cdot 3\cdot 3^3=4\cdot 27=108\text{ cm}^3\).

Answer: \(108\).

The volume of a cylinder is

\(V=\pi r^2h\).

If the radius is increased \(2\) times, then \(r^2\) increases by \(2^2=4\) times.

If the height is increased \(5\) times, the volume is additionally multiplied by \(5\).

Thus the total factor of increase is

\(4\cdot 5=20\).

Answer: \(20\).

As in the problem with a square axial section, the side of the square is

\(a=\dfrac{4\sqrt{2}}{\sqrt{2}}=4\text{ cm}\).

So

\(h=2r=4\), hence \(r=2\text{ cm}\), \(h=4\text{ cm}\).

The volume of the cylinder is

\(V=\pi r^2h\).

For \(\pi=3\):

\(V=3\cdot 2^2\cdot 4=3\cdot 4\cdot 4=48\text{ cm}^3\).

Answer: \(48\).

Using the formula for the base area of a cylinder,

\(\pi r^2=16\pi\).

So \(r^2=16\), hence \(r=4\text{ cm}\).

The axial section is a rectangle with sides \(2r\) and \(h\).

Its area is

\(S=2r\cdot h=2\cdot 4\cdot 4=32\text{ cm}^2\).

Answer: \(32\).

If a sphere is inscribed in a cube, then its diameter equals the edge of the cube:

\(d=10\text{ cm}\), so \(r=5\text{ cm}\).

The surface area of a sphere is

\(S=4\pi r^2\).

For \(\pi=3\):

\(S=4\cdot 3\cdot 5^2=12\cdot 25=300\text{ cm}^2\).

Answer: \(300\).

A plane section of a sphere is a circle. Its radius is found from

\(r^2=R^2-d^2\), where \(R=8\), \(d=7\).

Thus

\(r^2=8^2-7^2=64-49=15\).

The area of the circle is

\(S=\pi r^2=\pi\cdot 15\).

For \(\pi=3\):

\(S=45\text{ cm}^2\).

Answer: \(45\).

If a sphere is inscribed in a cylinder, then

\(r_{\text{cyl}}=r_{\text{sphere}}=2\text{ cm}\),

and the height of the cylinder equals the diameter of the sphere:

\(h=4\text{ cm}\).

The total surface area of the cylinder is

\(S_{\text{tot}}=2\pi r^2+2\pi rh=2\pi r(r+h)\).

Substitute the values and \(\pi=3\):

\(S_{\text{tot}}=2\cdot 3\cdot 2\cdot (2+4)=72\text{ cm}^2\).

Answer: \(72\).

For a right cylinder, the generatrix equals the height, so

\(h=6\text{ cm}\).

A section parallel to the axis is a rectangle. Its width equals the chord of the base at distance \(8\text{ cm}\) from the center.

Half of this chord is

\(x=\sqrt{10^2-8^2}=\sqrt{100-64}=6\text{ cm}\).

Therefore the whole chord is \(12\text{ cm}\).

Hence the area of the section is

\(S=12\cdot 6=72\text{ cm}^2\).

Answer: \(72\).

Such a sphere is circumscribed about the cube, so its radius equals half the space diagonal of the cube.

The space diagonal of the cube is

\(d=a\sqrt{3}=4\sqrt{3}\cdot \sqrt{3}=12\text{ cm}\).

Thus the sphere radius is

\(R=\dfrac{12}{2}=6\text{ cm}\).

The surface area of the sphere is

\(S=4\pi R^2\).

For \(\pi=3\):

\(S=4\cdot 3\cdot 6^2=12\cdot 36=432\text{ cm}^2\).

Answer: \(432\).

The volume of a cylinder is

\(V=\pi r^2h\).

If the radius is reduced by a factor of \(3\), then the new radius is \(\dfrac{r}{3}\), and the square of the radius decreases by a factor of \(9\):

\(\left(\dfrac{r}{3}\right)^2=\dfrac{r^2}{9}\).

To keep the volume unchanged, the height must compensate for this by increasing \(9\) times.

Answer: \(9\).