The diagonal of a regular 4-sided prism is 8√2 and is inclined to the base at an angle of 60 degrees. Find the area of the base.

Consider a right-angled triangle ACC1, whose angle C is straight, and the angle A, by condition, is 60, then the angle CC1A = 180 – 90 – 60 = 30.

The AC leg lies opposite angle 30, therefore, its length is equal to half the length of the AC1 hypotenuse.

AC = AC1 / 2 = 8 * √2 / 2 = 4 * √2 cm.

By condition, the pyramid is correct, therefore the base of AВСD is square, then AB = BC = СD = AD.

Consider an isosceles right-angled triangle ABC and, by the Pythagorean theorem, determine the lengths of the legs AB and BC.

AC ^ 2 = AB ^ 2 + BC ^ 2 = 2 * BC ^ 2.

(4 * √2) 2 = 2 * BC ^ 2.

32 = 2 * BC ^ 2.

BC ^ 2 = 32/2 = 16.

BC = 4 cm.

Then the base area is equal to:

Sbn = AB * BC = 4 * 4 = 16 cm2.

Answer: The base area is 16 cm2.