The surface area of the ball is 5n. The ball is cut by a plane. The circumference of the ball’s section is n.

The surface area of the ball is 5n. The ball is cut by a plane. The circumference of the ball’s section is n. Find the distance from the center of the ball to the secant plane?

The surface area of ​​the ball is determined by the formula:

Sш = 4пR ^ 2.

Knowing that the surface area of ​​the ball is 5n, we can find its radius:

R2 = Sш / 4п = 5п / 4п = 5/4;

R = √5 / 2 is the radius of this ball.

Knowing that the circumference of the section is equal to n, we can find the radius of the section:

l = 2пr;

r = l / 2п = п / 2п = 1/2.

Consider a right-angled triangle, in which the hypotenuse is the radius of the given ball, the legs are the radius of the section and the required distance m from the center of the ball to the cutting plane.

The distance m can be found by the Pythagorean theorem:

m ^ 2 = R ^ 2 – r ^ 2;

m ^ 2 = 5/4 – 1/4 = 4/4 = 1;

m = 1 is the distance from the center of the ball to the secant plane.