A spherical sector with an angle alpha in the axial section is highlighted in a ball of radius R. Find its volume.

univerkov | April 25, 2021 | education

The volume of the spherical sector is calculated by the formula: Vsec = 2 * n * R ^ 2 * H / 3, where R is the radius of the sphere, H is the height of the spherical segment, H = AH.

Let’s draw the HV radius at the base of the cone of the spherical segment. The height OH is the bisector of the angle at the vertex of the axial section of the spherical sector. Angle BON = BOC / 2 = α / 2.

Then in the triangle BOH, Cos (α / 2) = OH / OB = OH / R.

OH = R * Cos (α / 2).

Then the height of the spherical sector AH = R – R * Cos (α / 2) = R * (1 – Cos (α / 2)).

Then V = 2 * n * R ^ 2 * R * (1 – Cos (α / 2) / 3 = 2 * n * R ^ 3 * (1 – Cos (α / 2) / 3 cm3.

Answer: The volume of the spherical sector is 2 * n * R ^ 3 * (1 – Cos (α / 2) / 3 cm3.

One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.