# 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)).