The angle at the apex of the axial section of the cone is 90 degrees, the radius of the ball

| August 11, 2021 | education

The angle at the apex of the axial section of the cone is 90 degrees, the radius of the ball inscribed in the cone is 3√2 – 3. The volume of the cone is?

Consider the axial section of a cone and a ball inscribed in it.

The axial section of the cone is a right-angled isosceles triangle ABC, with an inscribed circle with center O and radius R = 3 * √2 – 3 cm.

Let’s build the radii OH, OK and OM. The OKBM quadrangle is a square, then its diagonal OB is equal to: OK * √2 = (3 * √2 – 3) * √2 = 6 – 3 * √2 cm.

Height BH = BO + OH = 6 – 3 * √2 + 3 * √2 – 3 = 3 cm.

The ABH triangle is rectangular and isosceles, since the angle ABH = 45, then AH = BH = 3 cm.

Determine the volume of the cone.

V = π * AH ^ 2 * BH / 2 = π * 9 * 3/3 = 9 * π cm3.

Answer: The volume of the cone is 9 * π cm3.



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