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

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: a) the side edge and the side of the base of the pyramid, b) the side surface of the pyramid, c) the full 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 DHS is rectangular, then CD ^ 2 = DH ^ 2 + CH ^ 2 = 225 + 243 = 468.

СD = 6 * √13 cm.

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

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

Determine the area of ​​the base of the pyramid. Sb = SV * AN / 2 = 18 * √3 * 27/2 = 243 * √3 cm2.

Then Sпов = Sbok + Sbn = 405 * √3 + 243 * √3 = 648 * √3 cm2.

Answer: The side edge is 6 * √13 cm, the side area is 405 * √3 cm2, the total surface is 648 * √3 cm2.

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