# The apothem of a regular triangular pyramid is 15 cm, and the segment connecting the top of the pyramid with

**The apothem of a regular triangular pyramid is 15 cm, and the segment connecting the top of the pyramid with the center of the base is -12 cm. Find the side surface of the pyramid**

The segment connecting the top of the pyramid with the center of the base is the height of the pyramid, then the triangle DOH is rectangular.

OH ^ 2 = DH ^ 2 – DO ^ 2 = 225 – 144 = 81.

OH = 9 cm.

AH in a regular triangle ABC is the height and median, then by the property of the medians, OA = 2 * OH = 2 * 9 = 18 cm.Then AH = OA + OH = 18 + 9 = 27 cm.

In a right-angled triangle AHC CH = AC / 2. Let CH = X cm, then AC = 2 * X cm.

By the Pythagorean theorem, 4 * X ^ 2 = AH ^ 2 + X ^ 2.

3 * X ^ 2 = 729.

X ^ 2 = 729/3 = 243.

X = CH = 9 * √3 cm.

CB = 2 * CH = 18 * √3 cm.

The triangle DCH is rectangular, then CD ^ 2 = DH ^ 2 + CH ^ 2 = 225 + 243 = 468.

СD = 6 * √13 cm.

Sdvs = CB * DH / 2 = 18 * √3 * 15/2 = 135 * √3 cm2.

Then Side = 3 * Sdvs = 3 * 135 * √3 = 405 * √3 cm2.

Answer: The side area is 405 * √3 cm2.