The cross section of a ball with an area of 16 π cm2 is located at a distance of 3 cm from the center

The cross section of a ball with an area of 16 π cm2 is located at a distance of 3 cm from the center of the ball. Find the surface area of the ball.

The section of the ball is a circle, the area of which is Ssection = πr ^ 2, where r is the radius of the section. By condition, the cross-sectional area of the ball is 16π cm2, which means:

πr ^ 2 = 16π;

r ^ 2 = 16;

r = √16 = 4 cm.

From a right-angled triangle formed by the radius r of a given section, the radius of the ball R and the perpendicular l drawn from the center of the ball to the plane, equal to 3 cm, according to the Pythagorean theorem, we can find the radius of the ball:

R ^ 2 = r ^ 2 + ^ l2 = 4 ^ 2 + 3 ^ 2 = 16 + 9 = 25;

R = √25 = 5 cm.

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

S = 4πR ^ 2 = 4 * π * 5 ^ 2 = 100π ≈ 314.16 cm2.