At the base of the prism ABCA1B1C1 is an equilateral triangle ABC with side a. The vertex A1 is projected to the center of the base

| August 7, 2021 | education

At the base of the prism ABCA1B1C1 is an equilateral triangle ABC with side a. The vertex A1 is projected to the center of the base, and the edge АА1 forms an angle with the base. Find the volume of a prism

Let us determine the length of the height AH of an equilateral triangle at the base of the prism.

AH = a * √3 / 2.

The heights in an equilateral triangle, at the point of intersection, are divided by a ratio of 2/1, starting at the apex.

AO = 2 * OH.

AO + OH = AH = AO + AO / 2 = 3 * AO / 2 = a * √3 / 2.

AO = a * √3 / 3 cm.

In the right-angled triangle AA1O, we define the leg OA1.

ОА1 = AO * tgα = (a * √3 / 3) * tanα cm.

The area of an equilateral triangle is equal to: Sax = a ^ 2 * √3 / 4 cm2.

Let’s define the volume of the prism.

V = Sosn * ОА1 = (a ^ 2 * √3 / 4) * (a * √3 / 3) * tanα = a ^ 3 * tanφ / 4 cm3.

Answer: The volume of the prism is a ^ 3 * tgφ / 4 cm3.



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