The face of a clock has a circumference of 63 inches. What is the area of the clock?
The Correct Answer and Explanation is :
To find the area of a clock given its circumference, we first need to understand the relationship between the circumference and the radius of a circle.
Step 1: Calculate the Radius
The formula for the circumference (C) of a circle is given by:
[
C = 2\pi r
]
where (r) is the radius of the circle. We can rearrange this formula to solve for the radius:
[
r = \frac{C}{2\pi}
]
Given that the circumference (C) of the clock is 63 inches, we can substitute this value into the formula:
[
r = \frac{63}{2\pi}
]
Using the approximation (\pi \approx 3.14):
[
r \approx \frac{63}{2 \times 3.14} \approx \frac{63}{6.28} \approx 10.03 \text{ inches}
]
Step 2: Calculate the Area
The area (A) of a circle is calculated using the formula:
[
A = \pi r^2
]
Now, we can substitute the value of (r) into this formula:
[
A \approx \pi (10.03)^2
]
Calculating (10.03^2):
[
10.03^2 \approx 100.6009
]
Now substituting back to find the area:
[
A \approx \pi \times 100.6009 \approx 3.14 \times 100.6009 \approx 318.34 \text{ square inches}
]
Conclusion
Thus, the area of the clock is approximately 318.34 square inches.
Explanation
This calculation shows how we derive the area of a circular object, like a clock face, starting from its circumference. The circumference provides a direct relationship to the radius, which is crucial since the area depends on the square of the radius. Understanding these relationships is fundamental in geometry, particularly in dealing with circular shapes. By employing formulas systematically, we can easily transition from linear measurements (like circumference) to area calculations. This process is valuable not only in mathematics but also in practical applications, such as design and manufacturing, where accurate area measurements are necessary.