The base of the pyramid is a rectangle with sides 6 and 8. Find the height of the pyramid if each side edge is 5 √5.

Since all the side faces of the pyramid are equal, its vertex is projected to the center of the circle described around the rectangle, that is, to the point of intersection of the diagonals.

In a right-angled triangle ACD, according to the Pythagorean theorem, we determine the length of the hypotenuse AC.

AC ^ 2 = AD ^ 2 + CD ^ 2 = 64 + 36 = 100.

AC = 10 cm.

The diagonals of the rectangle are equal and are divided in half at the intersection point, then OC = AC / 2 = 10/2 = 5 cm.

In a right-angled triangle KOC, by the Pythagorean theorem, we determine the length of the leg OK.

OK ^ 2 = CK ^ 2 – OC ^ 2 = 125 – 25 = 100.

OK = 10 cm.

Answer: The height of the pyramid is 10 cm.