The students measure another ball with the ruler in Problem \(2.93\) and determine that its diameter is \(2.55 \mathrm{~cm}\). What is the radius of the ball to the correct number of significant figures?

In scientific and mathematical contexts, significant figures relate to the precision of measurements and calculations. These are the digits in a number that carry meaningful information about its precision. For instance, in the diameter measurement of the ball, the value 2.55 cm has three significant figures, indicating that the precision extends to the hundredths of a centimeter.

One must be careful to carry through the correct number of significant figures throughout computations to ensure accuracy is maintained. When converting the diameter to the radius, the result should also display three significant figures as per the initial measurement. It is a common mistake to either round off numbers prematurely or to include more digits than necessary, both of which can lead to erroneous conclusions. Always check the original data for the number of significant figures and carry that precision through to your final answer.

The fundamental relationship within circle geometry is that the diameter is twice the length of the radius. If you know the diameter, you can find the radius by simply dividing by two. This diameter to radius conversion is essential in many areas of math and science.

To put the formula into action, simply follow this equation:
\[r = \frac{d}{2}\]
where \(r\) is the radius, and \(d\) is the diameter. In the context of our exercise, given a diameter of 2.55 cm, we apply the formula to find the radius. Division is straightforward, and because this step is a basic arithmetic operation, no additional significant figures are introduced by it. Therefore, the calculated radius must reflect the same level of precision as the original diameter measurement.

The study of circle geometry involves understanding the various properties and formulas related to a circle. These include the concepts of diameter, radius, circumference, and area, among others. One key formula, as we've explored, is the diameter-to-radius conversion. The radius is the distance from the center of the circle to any point on its circumference, and it plays a role in many circle-related calculations such as the area \(A = \pi r^2\) and the circumference \(C = 2\pi r\).

The geometry of a circle is fundamental in many fields, from engineering to astronomy. When we discuss the 'geometry' of a circle in the context of a problem, we are applying these ubiquitous relationships to find our desired measurements, whether it be the radius, diameter, or another quantity. It's the understanding and application of these relationships that enable us to solve for a circle's various aspects efficiently and accurately.