The height of the base of a regular triangular pyramid is 5 cm

| August 22, 2021 | education

The height of the base of a regular triangular pyramid is 5 cm, and the dihedral angle at the side of the base is 30 degrees. Find the area of the side surface.

The height AH of an equilateral triangle ABC is also its median, which at point O is divided in the ratio of 2 / 1. Then the length of the segment OH = 5/3 cm.

The DOН triangle is rectangular, then Cos30 = OH / DН.

DN = OH / Cos30 = (5/3) / (√3 / 2) = 10/3 * √3 cm.

Let the side length BC = X cm, then CH = X / 2 cm.

In a right-angled triangle ACН, according to the Pythagorean theorem, AH ^ 2 = AC ^ 2 – CH ^ 2.

25 = X ^ 2 – X ^ 2/4 = 3 * X ^ 2/4.

X ^ 2 = 4 * 25/3 = 100/3.

X = BC = 10 / √3 cm.

Determine the area of ​​the triangle DBC.

Sдвс = ВС * DH / 2 = (10 / √3) * (10/3 * √3) / 2 = 50/9.

Then Side = 3 * Sdvs = 3 * 50/9 = 50/3 = 16 (2/3) cm2.

Answer: The lateral surface area is 16 (2/3) cm2.



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