Final answer:
The diameter of a circle is twice the radius, and the perimeter (circumference) is calculated using the formula 2πr, where r is the radius. The radius is half of the circle's diameter, and the circumference is always less than the perimeter of the containing square but more than the direct distance across it.
Step-by-step explanation:
For a given circle, the diameter is the longest straight line that can be drawn from one side of the circle to the other, passing through the center. The radius is half of the diameter and is the distance from the center of the circle to any point on its perimeter. Now, the perimeter of a circle, also known as the circumference, is the total distance around the circle. Given that the diameter (d) is twice the radius (2r), and the side length of a square (a) is equal to the diameter (2r), the perimeter of the circle can be expressed as 2πr, where π (pi) is approximately 3.14159. Since the perimeter must be greater than the direct round trip across the square (2a) but less than the perimeter of the square (4a), it makes sense to find that the circle's perimeter is approximately 3 times the side of the square or 6 times the radius, leading to the formula 2πr for the circumference.
To remember which formula goes with which measurement, it's helpful to think about units. The perimeter or the circumference has units of length, such as meters, whereas the area formula (πr²) contains squared units, such as square meters, making it easy to distinguish between the two.