- Start with the formula for the area of a sector: \(A = \frac{\pi r^2 S}{360}\).
- Multiply both sides by 360 to get \(360 A = \pi r^2 S\).
- Divide by \(\pi r^2\) to isolate the angle \(S\), resulting in \(S = \frac{360 A}{\pi r^2}\).
- The final expression is \(\boxed{\frac{360 A}{\pi r^2}}\).
Explanation
- Understanding the Problem
The area of a sector of a circle is given by the formula \(A = \frac{\pi r^2 S}{360}\), where \(A\) is the area, \(r\) is the radius, and \(S\) is the central angle of the sector in degrees. Our goal is to isolate the variable \(S\) to find its expression in terms of the other variables.
- Solving for \(S\)
To isolate \(S\), we begin with the original equation:
\[A = \frac{\pi r^2 S}{360}\]
First, multiply both sides of the equation by \(360\) to clear the fraction:
\[360 \times A = \pi r^2 S\]
Next, divide both sides of the equation by the term \(\pi r^2\) to isolate \(S\):
\[S = \frac{360 A}{\pi r^2}\]
Thus, solving for \(S\) gives us the expression \(\frac{360 A}{\pi r^2}\).
- Final Conclusion
Comparing our derived result with the given choices, we see that the correct equation is the fourth option: \(\frac{360 A}{\pi r^2}\). Thus, the correct expression Mia wrote is \(\boxed{\frac{360 A}{\pi r^2}}\).
Examples
In a real-world scenario, such as landscape design or architecture, you might need to determine the central angle for a circular garden bed or window arch. If you know the total surface area you want to cover and the radius of the space, you can use the sector area formula to calculate the exact angle required to achieve your design, ensuring precision in your construction. Rearranging the formula as we did here makes it simple to solve for the angle when area and radius are known variables.