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: \(6\).