The diameter of a circle is 8 inches. What is the circumference?
The Correct Answer and Explanation is:
The circumference of the circle can be calculated using the formula:
C=π×dC = \pi \times d
Substituting the given diameter of 8 inches:
C=π×8=8πC = \pi \times 8 = 8\pi
Thus, the exact circumference is 8π inches.
Explanation:
The circumference represents the total length of the outer boundary of a circle. This property is fundamental in geometry and finds applications in numerous fields, including engineering, architecture, and everyday measurements.
The formula for the circumference stems from the relationship between a circle’s diameter and the mathematical constant π (pi). The number π is an irrational constant that approximately equals 3.14159, though maintaining it in symbolic form ensures precision in calculations. Multiplying π by the diameter provides the exact circumference.
Understanding the significance of this formula helps in various real-world situations. When designing circular objects such as wheels, rings, and round tables, the circumference determines material requirements and dimensional accuracy. In sports, the circumference of running tracks or circular fields plays a crucial role in calculating distances. Even in physics, wave propagation and circular motion problems often require precise circumference calculations.
Instead of using approximate decimal values, the answer remains in terms of π, ensuring exactness. This approach proves useful when working with advanced mathematical concepts, where unnecessary rounding can lead to errors.
Fundamentally, circles are among the most symmetric and essential shapes studied in mathematics. The ability to compute their circumference reinforces deeper comprehension of geometric properties, paving the way for applications in science, engineering, and technology. Through proper analysis, fundamental relationships such as the connection between diameter, radius, and circumference become clearer, strengthening proficiency in problem-solving.
