The apothem of a regular triangular pyramid SABC is 10. The side of the base is 12. What is the height of the pyramid?

Let’s draw the height of the triangle at the base of the pyramid. Since triangle ABC is equilateral, its height AH is also the median and bisector of the triangle, then CH = BH = BC / 2 = 12/2 = 6 cm.In a right-angled triangle ACН, according to the Pythagorean theorem, AH ^ 2 = AC ^ 2 – CH ^ 2 = 144 – 36 = 108.

AH = 6 * √3 cm.

The medians of the triangle, at the intersection point, are divided by a ratio of 2/1, starting at the apex.

OA / OH = 2/1.

OA = 2 * OH.

AH = OA + OH = 3 * OH = 6 * √3.

OH = 6 * √3 / 2 = 3 * √3.

In a right-angled triangle DOН, according to the Pythagorean theorem, OD ^ 2 = DН ^ 2 – OH ^ 2 = 100 – 27 = 73.

OD = √73 cm.

Answer: The height of the pyramid is 73 cm.