Given a regular triangular pyramid, the side of the base of which is 12 cm, its lateral edge makes an angle

| September 3, 2021 | education

Given a regular triangular pyramid, the side of the base of which is 12 cm, its lateral edge makes an angle of 45 degrees with the base. Find V and S complete?

In an equilateral triangle ABC, draw the height AH and determine its length.

AH = a * √3 / 2, where a is the length of the side of the triangle. AH = 12 * √3 / 2 = 6 * √3 cm.

Determine the area of ​​the base of the pyramid. Sbn = ВС * АН / 2 = 12 * 6 * √3 / 2 = 36 * √3 cm2.

The segment AO is the radius of a circle circumscribed around the triangle ABC, then AO = BC / √3 = 12 / √3 = 4 * √3 cm.

The DAO triangle is rectangular and isosceles, since its angle A is 450, then DO = AO = 4 * √3 cm.

Let’s define the volume of the pyramid.

V = Sbn * DO / 3 = (36 * √3 * 4 * √3) / 3 = 144 cm3.

Let’s make an apothem of DN. By the property of the heights of an equilateral triangle OH = AO / 2 = 4 * √3 / 2 = 2 * √3 cm.Then DN ^ 2 = OH ^ 2 + DO ^ 2 = 12 + 48 = 60.

DN = 2 * √15 cm.

Determine the area of ​​the side face. Svsd = ВС * ДН / 2 = 12 * 2 * √15 / 2 = 12 * √15, then

Sside = 3 * Svsd = 3 * 12 * √15 = 36 * √15 cm2.

S floor = S main + S side = 36 * √3 + 36 * √15 = 36 * √3 * (1 + √5) cm2.

Answer: The volume of the pyramid is 144 cm3, the surface area is 36 * √3 * (1 + √5) cm2.



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