# The height of a regular quadrangular prism is 4 cm, the diagonal of the prism is √24 cm.

**The height of a regular quadrangular prism is 4 cm, the diagonal of the prism is √24 cm. Determine the area of the lateral surface of the prism.**

According to the condition, the prism is correct, therefore the angle C1CA = 90, AB = BC = CD = AD.

Consider a right-angled triangle AC1C, in which the hypotenuse AC1 = √24 cm, leg CC1 = 4 cm, then, according to the Pythagorean theorem, the leg AC will be equal to:

AC = √ (AC1 ^ 2 – CC1 ^ 2) = √ (24 ^ 2 – 4 ^ 2) = √ (24 – 16) = √8 = 2 * √2 cm.

Consider a right-angled triangle ACD, in which AD = CD, and AC = 2 * √2 cm.

Then, by the Pythagorean theorem, AC ^ 2 = AD ^ 2 + CD ^ 2 = 2 * AD ^ 2.

АD = (√АС ^ 2) / √2 = √ (2 * √2) ^ 2 / √2 = 2 cm.

Let us determine the area of the lateral surface of the prism:

Side = 4 * AD * CC1 = 4 * 2 * 4 = 32 cm2.

Answer: S side = 32 cm2.