# 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.

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