The lateral edge of a regular triangular pyramid is 3 times the edge of the base, and the sum of the lengths of all the edges

| August 22, 2021 | education

The lateral edge of a regular triangular pyramid is 3 times the edge of the base, and the sum of the lengths of all the edges of the pyramid is 36. Find the length of the apothem of the pyramid.

Let the length of the rib at the base of the pyramid be X cm, then the length of the side rib is 3 * X cm.

Since, by condition, the sum of the lengths of all edges is 36 cm, then 3 * (X + 3 * X) = 36.

12 * X = 36.

X = 36/12 = 3.

AB = BC = AC = 3 cm.

DА = DВ = DC = 3 * 9 = 9 cm.

The DBC triangle is isosceles, then the apothem DH of the pyramid, so is the height and median of the DBC triangle.

Then ВН = СН = ВС / 2 = 3/2 cm.

In a right-angled triangle DH ^ 2 = DC ^ 2 – CH ^ 2 = 81 – 9/4 = 315/4.

DН = √315 / 2 = 3 * √35 / 2 cm.

Answer: The length of the apothem is 3 * √35 / 2 cm.



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