The lateral edge of a regular pyramid quadrilateral is 4 cm and makes an angle of 45 degrees with the base plane

The lateral edge of a regular pyramid quadrilateral is 4 cm and makes an angle of 45 degrees with the base plane of the pyramid. Find the height of the pyramid.

Since the pyramid is correct, there is a square at its base.

The vertex P of the pyramid PABCD is projected to point O, the point of intersection of the diagonals of the square.

In a right-angled triangle AOP, the angle OAP, by condition, is 45, then the triangle OAP is rectangular and isosceles, OA = OP.

Then AP ^ 2 = OA ^ 2 + OP ^ 2 = 2 * OP ^ 2.

OA ^ 2 = AP ^ 2/2 = 16/2 = 8.

OR = √8 = 2 * √2 cm.

Answer: The height of the pyramid is 2 * √2 cm.