The diagonal of a regular quadrangular prism is 22 cm, and the height is 14 cm. Find the total surface area of this prism.

The diagonal of the prism, its height and the diagonal of the base form a right-angled triangle. By the Pythagorean theorem, we find the square of the diagonal of the base as the difference between the squares of the diagonal of the prism and the height:

d ^ 2 = D ^ 2 – h ^ 2 = 22 ^ 2 – 14 ^ 2 = 484 – 196 = 288.

The total surface area of ​​the prism is equal to the sum of the lateral surface areas and the two bases:

S full = S side + 2 S main.

Since the prism is correct, there is a square at its base, the area of ​​which can be defined as half the value of the square of its diagonal:

Sb = d ^ 2/2 = 288/2 = 144 cm2.

On the other hand, the area of ​​a square is equal to the square of its side:

Sbn = a ^ 2.

Hence, the side of the base is equal to:

a = √Sbase = √144 = 12 cm.

The four side faces of this prism are equal rectangles, one of the sides of which is equal to the side of the base, the other is the height of the prism. The area of ​​such a rectangle is:

S side.gr = a * h = 12 * 14 = 168 cm2.

Find the total surface area:

Sful = S side + 2Sb = 4 * S side.gr + 2 * Sb = 4 * 168 + 2 * 144 = 960 cm2.