The circumference of a circle is \( 6 \pi \mathrm{ft} \). What is the area, in square feet? Expres your answer in terms of \( \pi \).

To find the area of the circle, we first need to determine the radius. The circumference \( C \) of a circle is given by the formula \( C = 2\pi r \). Given that the circumference is \( 6\pi \) feet, we can set up the equation: \[ 2\pi r = 6\pi \] Dividing both sides by \( 2\pi \): \[ r = \frac{6\pi}{2\pi} = 3 \text{ feet} \] Now, we use the formula for the area \( A \) of a circle, which is given by \( A = \pi r^2 \): \[ A = \pi (3)^2 = \pi \times 9 = 9\pi \text{ square feet} \] So, the area of the circle is \( 9\pi \) square feet. Speaking of circles, did you know that circles have fascinated humans throughout history? The ancient Greeks studying conic sections laid the groundwork for our understanding of geometry today. Mathematicians like Archimedes even devised famous methods for calculating areas of circular shapes, leading to the concept of limits that plays a crucial role in calculus! Now, if you ever need to apply this knowledge practically, consider how the area of circular gardens or ponds can inform landscaping decisions. Knowing the area helps in estimating how much mulch or plant cover you’d need. Avoid the common mistake of forgetting to square the radius when calculating area—it’s an easy slip that can lead to significantly underestimating the materials needed!