In the cone, through its apex at an angle φ to the base plane, a plane is drawn, cutting off the arc 2α from the base circle.
| September 21, 2021 | education
In the cone, through its apex at an angle φ to the base plane, a plane is drawn, cutting off the arc 2α from the base circle. The radius of the base of the cone is R. Find the volume of the cone.
By condition, the arc MK = 2α, then the central angle MOK = 2α.
The MOC triangle is isosceles, since OM = OK = R, then the OH height is the median and the bisector. Then the angle МОН = 2α / 2 = α.
In the MON triangle, Cosα = OH / OM = OH / R.
OH = R * Cosα see.
In a right-angled triangle BOH, tg φ = OB / OH.
OB = OH * tan φ = R * Cosα * tan φ.
Determine the volume of the cone.
V = π * R ^ 2 * OB / 3 = π * R ^ 2 * R * Cosα * tanφ / 3 = π * R ^ 3 * Cosα * tanφ / 3 cm3.
Answer: The volume of the cone is π * R ^ 3 * Cosα * tgφ / 3 cm3.
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