Regular quadrangular prisms are described near the cylinder, the diagonal of which

Regular quadrangular prisms are described near the cylinder, the diagonal of which is equal to d. Side diagonal = b. Find the volume of the cylinder.

From the right-angled triangle DA1B1, about the Pythagorean theorem, A1B1 = √ (d ^ 2 – b ^ 2) see.

The radius of the inscribed cylinder is equal to half the length of A1B1.

R = √ (d ^ 2 – b ^ 2) / 2 cm.

Determine the area of the base of the cylinder. Sop = π * R ^ 2 = π * (d ^ 2 – b ^ 2) / 2 cm2.

Let us determine the length of the diagonal BD.

BD ^ 2 = AB ^ 2 + AD ^ 2 = 2 * (d ^ 2 – b ^ 2).

In a right-angled triangle DVB1, according to the Pythagorean theorem, BB1 ^ 2 = DV1 ^ 2 – VD ^ 2 = d ^ 2 – 2 * d ^ 2 + 2 * b ^ 2 = 2 * b ^ 2 – d ^ 2.

BB1 = √ (2 * b2 – d ^ 2).

Then V = Sax * BB1 = π * (d ^ 2 – b ^ 2) * √ (2 * b ^ 2 – d ^ 2) / 2 cm3.

Answer: The volume of the cylinder is π * (d ^ 2 – b ^ 2) * √ (2 * b ^ 2 – d ^ 2) / 2 cm3.