Final answer:
The area of a circle is reduced to one-fourth its original size when the radius is cut down to half, as the area is directly proportional to the square of the radius by the formula πr². Reducing the radius by half results in an area of π(r/2)², which is πr²/4, or one-fourth of the original area.
Step-by-step explanation:
When considering the effect on the area of a circle when its radius is cut down to half, we need to understand the relationship between the radius of a circle and its area. The area of a circle is calculated by the formula πr², where r represents the radius of the circle. If the radius is cut down to half, the new radius r' will be r/2. Plugging this into the area formula gives us π(r/2)²=πr²/4. Thus, the new area is one-fourth of the original area.
Using examples from similar principles, if you double the radius, you increase the area by a factor of 4, as evidenced when considering the gravitational force, which reduces by a factor of (1/2)² when the radius is doubled. Inversely, reducing the radius to one-half will effectively make the area one-fourth the size. This principle can be observed in various physical phenomena, such as illuminance decreasing with the inverse square of the distance, orbital velocity of satellites, or volume flow rates in physics.