In a regular quadrangular pyramid, the height is 3, the lateral edge is 6. Find the radius of the sphere described around the pyramid.

Let’s construct a section passing through the axis of the ball, the vertex M of the pyramid and the diagonal AC of the square at the base of the pyramid.

In the section, an isosceles triangle MAC is inscribed in a circle, in which the sides are 6 cm, and the height and median MO1 = 3 cm.

Then in a right-angled triangle AOM1 the angle against the leg MO1 is 300, and then the angle AMO1 = 90 – 30 = 60.

In the triangle AOM, OA = OM = R, and since the angle AMO = 60, then the triangle AOM is equilateral, and then R = AM = 6 cm.

Answer: The radius of the ball is 6 cm.