A regular quadrangular pyramid is given, apothem is 2a, height = a √2. Find the side of the base or the edge of the base.

Let’s draw the diagonals of the base of the pyramid and connect the point of their intersection O with the apothem KN.

In the formed right-angled triangle OKH, we determine the length of the OH leg.

OH ^ 2 = KH ^ 2 – KO ^ 2 = (2 * a) ^ 2 – (a * √2) ^ 2 = 4 * a ^ 2 – 2 * a ^ 2 = 2 * a ^ 2.

OH = a * √2 cm.

Since the CKD triangle is isosceles, the KH apothem is the median of the CKD triangle, which means it divides the base of the CD in half. CH = DH.

Since in a triangle ACD OA = OC, and CH = DH, then OH is the middle line of the triangle, and then AD = 2 * OH = 2 * a * √2 cm.

Answer: The side of the base is 2 * a * √2 cm.