In a regular quadrangular pyramid, the height is 12 cm, and the apothem is 15 cm. Find the length of the base rib.

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 = 225 – 144 = 81.

OH = √81 = 9 cm.

The OH segment is the middle line of the triangle ABC, then AB = 2 * OH = 2 * 9 = 18 cm.

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

AC ^ 2 = 2 * AB ^ 2 = 2 * 324 = 648.

AC = √648 = 18 * √2.

Then OS = 18 * √2 / 2 = 9 * √2 cm.

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

PC ^ 2 = PO ^ 2 + OS ^ 2 = 144 + 162 = 306.

РС = √306 = 3 * √34 cm.

Answer: The length of the side is 3 * √34 cm.