Final answer:
Doubling the diameter or radius of a circle doubles the circumference and quadruples the area. Halving the diameter halves the circumference and reduces the area to one quarter of its original size.
Step-by-step explanation:
The effect of changes in a circle's diameter or radius on its area and circumference can be explained using the formulas for circumference (C = 2πr) and area (A = πr²), where r is the radius of the circle.
- (a) Doubling the diameter of a circle effectively doubles its radius since the diameter is twice the radius. This results in the circumference becoming twice as large because the circumference formula has a linear relationship with radius. For the area, doubling the radius squares the value within the area formula, yielding four times the original area. Thus, doubling the diameter results in the circumference doubling and the area increasing by four times.
- (b) Halving the diameter means the radius is also halved. The circumference becomes half its original length, and the area becomes one-quarter of its original value since the area is proportional to the square of the radius.
- (c) Doubling the radius has the same effect as doubling the diameter. The circumference will double, and the area will increase by a factor of four.
When considering these changes, it's crucial to remember that the circumference depends linearly on the radius, while the area depends quadratically.