Final answer:
Halving the radius of a sphere results in the surface area becoming one-quarter of its original value due to the quadratic relationship between radius and surface area, encapsulated by the formula 4πr2.
Step-by-step explanation:
When the radius of a sphere is halved, the surface area of the sphere decreases to one-quarter of the original surface area. The surface area of a sphere is given by the formula 4πr2, where r is the radius. Therefore, if the radius is halved, the new surface area becomes 4π(½ r)2 = πr2, which is one-quarter of the initial surface area since (1/2)2 equals 1/4.
This principle involves the square of the radius, meaning any change to the radius will affect the surface area by the square of that change. It follows the broader mathematical concept that any geometric property dependent on the radius squared will undergo a quadratically proportional change when the radius is altered.