The height of a regular quadrangular pyramid is 6, and the apothem is 6.5. Find the perimeter of the base of this pyramid

Let’s make an apothem of the PH of the pyramid. The side faces of the pyramid are isosceles triangles, then PH is the height, median and bisector of triangle ABP, then AH = BH.

At the base of the pyramids lies a square, then the triangle ABC is rectangular and isosceles, and CO = AO as half of the AC diagonal. Then OH is the middle line of the triangle ABC. OH = AB / 2.

From the right-angled triangle PHO, we determine the length of the leg OH.

OH ^ 2 = PH ^ 2 – PO ^ 2 = 42.25 – 36 = 6.25.

OH = 2.5 cm.

Then AB = 2 * 2.5 = 5 cm.

Let’s define the perimeter of the base.

Ravsd = 4 * AB = 4 * 5 = 20 cm.

Answer: The perimeter of the base is 20 cm.