Find the volume of a regular quadrangular prism inscribed in a cylinder whose base radius

Find the volume of a regular quadrangular prism inscribed in a cylinder whose base radius is equal to the root of 8 and the height is 7.

Given:
Right. 4-corner. prism;
Prism inscribed into the cylinder;
R = √8;
h-7;
V prism =?
Solution:
1) Because according to the condition of the problem, the prism is regular quadrangular, therefore the base of the prism is a square.
Because the prism is inscribed in the cylinder, which means that the base of the prism is inscribed in the base of the cylinder, i.e. we have a square inscribed in a circle. There is the following ratio between the side of the inscribed square and the radius of the circle:
а₄ = R√2,
where a₄ is the side of the inscribed square;
R is the radius of the circle.
Then a₄ = √8 * √2 = √16 = 4;
2) The volume of the prism is equal to the product of the base area by the height:
V prisms = Sh;
Because at the base of the prism is a square, therefore the area of ​​the base is equal to the square of the side of the base.
S = a₄²;
S = 4² = 16;
Because the prism is inscribed in the cylinder, so the height of the prism is equal to the height of the cylinder.
Then the volume of the prism is:
V prism = 16 * 7 = 112
Answer: 112