The base of prism ABCA1B1C1 is an equilateral triangle. The vertex A1 is projected to the center of this base, the edge AA1

| September 5, 2021 | education

The base of prism ABCA1B1C1 is an equilateral triangle. The vertex A1 is projected to the center of this base, the edge AA1 makes an angle “phi” with the base plane, find the volume of the prism if its height is h.

Point O is the projection of the vertex A1 onto the plane of the base ABC. Since the bases of the prism are equilateral triangles, the point O is the point of intersection of the medians of the triangles, which means that the point O divides the median AM into segments AO and OM, the ratio of which is 2/1.

In a right-angled triangle AA1O, we determine the length of the leg AO.

tgOAA1 = tgφ = ОА1 / AO = h / AO.

AO = h / tgφ. Then AM = AO + OM = (h / tgφ) + (h / 2 * tgφ) = 3 * h / 2 * tgφ see.

The height of an equilateral triangle is: AM = BC * √3 / 2 = 3 * h / 2 * tgφ.

ВС = 3 * h / √3 * tgφ = h * √3 / tgφ see.

Then Sosn = AM * BC / 2 = (3 * h / 2 * tgφ) * (h * √3 / tgφ) / 2 = 3 * √3 * h2 / 4 * tg2φ =

3 * √3 * h ^ 2 * ctg2φ / 4 cm2.

Then the volume of the prism is:

V = Sbn * h = 3 * √3 * h ^ 2 * ctg2φ * h / 4 = h ^ 3 * 3 * √3 * ctg2φ / 4 cm3.

Answer: The volume of the prism is h3 * 3 * √3 * ctg2φ / 4 cm3.



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