The apothem of a regular quadrangular pyramid is twice the side of the base.

The apothem of a regular quadrangular pyramid is twice the side of the base. Find the tangent of the angle between the planes of the side face and the base of the pyramid.

Let the length of the side of the base of the pyramid be equal to X cm, then by condition, the apothem KH = 2 * X cm.

The OH segment is equal to half the length of the side of the base, since OH is the middle line of the ACD triangle. OH = AD / 2 = X / 2 cm.

In a right-angled triangle KHO, KO ^ 2 = KH ^ 2 – OH ^ 2 = 4 * X ^ 2 – X ^ 2/4 = 15 * 2/4.

KO = X * √15 / 2.

Then tgKHO = KO / OH = (X * √15 / 2) / (X / 2) = √15.

Answer: The tangent of the angle between the planes is √15.