In a ball with a radius of 13 cm, two equal parallel sections with a radius of 5 cm are drawn

In a ball with a radius of 13 cm, two equal parallel sections with a radius of 5 cm are drawn on opposite sides of the center. Find the volume of the resulting spherical layer.

Let’s connect the center of the ball with the center of the circle in the section. The segment OO1 is perpendicular to the section plane. In a right-angled triangle COO1, OC is equal to the radius of the ball, O1C to the radius of the section, then, according to the Pythagorean theorem, we determine the distance from the center of the ball to the section.

OO1 ^ 2 = OC ^ 2 – O1C ^ 2 = 169 – 25 = 144.

OO1 = 12 cm.Then O1D = OO – OO1 = 13 – 12 = 1 cm.

Let’s define the volume of the ball.

Vball = 4 * n * R3 / 3 = 4 * n * 133/3 = 8788 * n cm3.

Let’s define the volume of the spherical segment.

Vseg = n * O1D ^ 2 * (OD – O1D / 3) = n * 1 * (13 – 1/3) = 38 * n / 3 cm3.

Since the second segment is removed at the same distance, its volume is also 38 * p / 3 cm3.

Then Vlayer = Vball – 2 * Vseg = 8788 * n / 3 – 76 * n / 3 = 2904 * n cm3.

Answer: The volume of the spherical layer is 2904 * n cm3.