# An equilateral cylinder is inscribed in the cone. Find the height of the cylinder if the height

An equilateral cylinder is inscribed in the cone. Find the height of the cylinder if the height of the cone is H and the angle at the vertex of the axial section is alpha.

Let the height of the cylinder be OO1 = X cm. Since the axial section of the cylinder is square, BO1 = OO1 = X cm.

The length of the segment CO1 = CO – OO1 = h – X cm.

In a right-angled triangle AOC tg (α / 2) = AO / CO.

AO = CO * tg (α / 2) = h * tg (α / 2), then AE = 2 * h * tg (α / 2).

Triangles ACE and ВСD are similar in two angles, then:

О1С / ВD = СО / АЕ.

(h – X) / X = h / 2 * h * tan (α / 2).

X * h = 2 * h2 * tan (α / 2) – X * 2 * h * tan (α / 2).

X * h + X * 2 * h * tan (α / 2) = 2 * h2 * tan (α / 2).

X = (2 * h * tan (α / 2)) / (1 + 2 * tan (α / 2)).

Answer: The height of the cylinder is (2 * h * tan (α / 2)) / (1 + 2 * tan (α / 2)).

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