In a regular quadrangular pyramid, the height is 16 cm, and the height of the side face (apothem) is 21 cm. Find the side edge.

Let’s draw a segment OH between the center of the base and the end of the apothem. The POH triangle is rectangular, then, according to the Pythagorean theorem, OH ^ 2 = PH ^ 2 – OP ^ 2 = 441 – 256 = 185.

OH = √185 cm.

The OH segment is the middle line of the triangle ABC, then AB = 2 * OH = 2 * √185 cm.

Let us determine the length of the AC diagonal of the AВСD square.

AC ^ 2 = 2 * AB ^ 2 = 2 * 4 * 185 = 1480.

AC = √1480 = 2 * √370.

Then OС = 2 * √370 / 2 = √370 cm.

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

PC ^ 2 = PO ^ 2 + OC ^ 2 = 256 + 370 = 626.

РС = √626 cm.

Answer: The length of the side is √626 cm.