In a regular quadrangular prism, the side of the base is 3 cm, calculate the height of this prism if its diagonal is 12 cm.

At the base of a regular quadrangular prism is a square, then AB = BC = CD = AD = 3 cm.

Let us draw the diagonal of the base of AC and, by the Pythagorean theorem, determine the length of the diagonal of AC.

AC ^ 2 = AB ^ 2 + BC ^ 2 = 9 + 9 = 18.

AC = 3 * √2 cm.

From the right-angled triangle ACC1, using the Pythagorean theorem, we determine the length of the leg CC1.

CC1 ^ 2 = AC1 ^ 2 = AC ^ 2 = 144 – 18 = 126.

CC ^ 2 = √126 = 3 * √14 cm.

Answer: The height of the prism is 3 * √14 cm.