Question
The circumference of a circle is \( 12 \pi \mathrm{ft} \). What is the area, in square feet? Express your answer in terms of \( \pi \).
Ask by Pearson Weaver. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The area of the circle is \( 36\pi \) square feet.
Solution
Given the circumference of the circle is 12π feet, we can use the formula for the circumference of a circle, which is:
Circumference = 2πr
where r is the radius of the circle. Setting up the equation:
2πr = 12π
Divide both sides of the equation by 2π:
r = (12π) / (2π) = 6 ft
Now, we can find the area using the area formula for a circle:
Area = πr²
Substitute r = 6 ft into the formula:
Area = π(6)² = 36π ft²
So, the area of the circle is 36π square feet.
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Simplify this solution Extra Insights
To find the area of the circle, we first need to determine its radius from the given circumference. The formula for the circumference \( C \) of a circle is: \[ C = 2\pi r \] where \( r \) is the radius. We have \( C = 12\pi \). Setting the two expressions for circumference equal gives us: \[ 2\pi r = 12\pi \] Dividing both sides by \( 2\pi \): \[ r = \frac{12\pi}{2\pi} = 6 \text{ ft} \] Now, to find the area \( A \) of the circle, we use the formula: \[ A = \pi r^2 \] Substituting the radius: \[ A = \pi (6)^2 = \pi \cdot 36 = 36\pi \] Thus, the area of the circle is \( 36\pi \) square feet.
