Question
area of a circular ring with an as: \( A=\pi\left(R^{2}-r^{2}\right) \).
Ask by Thornton Williams. in South Africa
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
To find the area of a circular ring, use the formula \( A = \pi(R^2 - r^2) \). Identify the outer radius \( R \) and inner radius \( r \), square them, subtract \( r^2 \) from \( R^2 \), and multiply the result by \( \pi \).
Solution
To find the area of a circular ring using the formula \( A = \pi(R^2 - r^2) \), we need to identify the variables involved:
- \( R \): the radius of the outer circle.
- \( r \): the radius of the inner circle.
The area \( A \) represents the space between the two circles.
### Steps to Calculate the Area:
1. **Identify the Values**: Determine the values of \( R \) and \( r \).
2. **Calculate \( R^2 \) and \( r^2 \)**: Compute the squares of the outer and inner radii.
3. **Subtract \( r^2 \) from \( R^2 \)**: Find the difference between the two squares.
4. **Multiply by \( \pi \)**: Finally, multiply the result by \( \pi \) to get the area of the circular ring.
If you provide specific values for \( R \) and \( r \), I can perform the calculations for you.
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Simplify this solution Bonus Knowledge
Did you know that the formula for the area of a circular ring ties directly to the concept of "disk subtraction"? Essentially, you're finding the area of the larger circle and then taking away the area of the smaller circle inside it. So if you have a big pizza (with radius \( R \)) and a small pizza (with radius \( r \)), you're calculating the area of the crust—the part you might actually want to eat! When using the formula \( A=\pi\left(R^{2}-r^{2}\right) \), a common mistake is neglecting to square the radii before subtracting. Always make sure to calculate \( R^2 \) and \( r^2 \) first, otherwise you might end up with a crust that's either larger or smaller than you intended!
