Final answer:
To find a missing diameter of a circle, double the radius, and to find a missing radius, halve the diameter. The circumference is π times the diameter, or twice π times the radius. The area of a circle is π times the square of the radius.
Step-by-step explanation:
Understanding the relationship between the radius and diameter of circles is fundamental in Geometry. The diameter (d) of a circle is twice the length of its radius (r), and vice versa, the radius is half the length of its diameter. This concept can be used to fill in the missing values for the radius or the diameter in any table. Moreover, the circumference of a circle is related to the diameter by the formula C = πd, also expressed as C = 2πr because the diameter is twice the radius. When considering the area of a circle, the area is always less than the area of the square that contains it, calculated as A = πr².
When the relationship a = 2r is mentioned, it implies that the diameter of the circle matches the side of the square, thereby fitting the circle perfectly inside the square. According to the information provided, the perimeter of the circle is less than a square's perimeter but significantly more than 2a. Similarly, a circle's area inside a square is less than the square's area (a²), but more than half of it, likely around three-quarters of the square's area, thus close to πr².
When converting these geometric concepts to real-world measurements, such as the diameter of Earth, they help to provide tangible contexts and transform abstract numbers into comprehensible facts. Additionally, the method for finding atomic radii by halving the distance between the nuclei of diatomic molecules can be thought of as a microcosm of finding the radius from diameter (e.g., “F radius equals 64 pm” because F diameter is 128 pm). Whether dealing with macroscopic or microscopic scales, the relationship between radius, diameter, and circumference remains constant.