The base side of a regular hexagonal pyramid is 6 dm, the angle between the planes of the side face

| May 23, 2021 | education

The base side of a regular hexagonal pyramid is 6 dm, the angle between the planes of the side face and the base is 30 degrees. Calculate the length: 1) the apothem of the pyramid; 2) the height of the pyramid.

At the base of the pyramid, we construct the segments OA and OB, which are the radii of the circle inscribed in the base. In a regular hexagon, the radius of the inscribed circle is equal to the side of the hexagon, then OA = OB = AB = 6 cm, and then the triangle AOB is equilateral.

OH is the height of an equilateral triangle. OH = AB * √3 / 2 = 3 * √3 cm.

The OHP triangle is rectangular, in which PH is the apothem, OP is the height of the pyramid, the angle OHP = 30 by condition.

tg30 = OP / OH.

OR = OH * tg30 = 3 * √3 * √3 / 3 = 3 cm.

By the Pythagorean theorem, PH ^ 2 = OH ^ 2 + OP ^ 2 = 27 + 9 = 36.

PH = 6 cm.

Answer: The length of the apothem is 6 cm, the length of the height is 3 cm.



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