Determining the inner and outer radii of a ring (annulus) is straightforward if you know the area of the ring and the thickness of the ring. Follow this step-by-step guide to understand how to do it.
Step 1: Understand the Formula
To find the inner and outer radii, you will use the given area of the ring and the thickness. The formulas you will use are:
\[ \text{Area of Ring} = \pi R^2 - \pi r^2 \]
\[ \text{Thickness} = R - r \]
where:
- \( R \) is the outer radius
- \( r \) is the inner radius
- \( \pi \) (Pi) is approximately 3.14159
Step 2: Use Real Numbers for Calculation
Let's use a real example to make this clear. Suppose the area of the ring is 150 square units and the thickness of the ring is 3 units.
Step 3: Set Up the Equations
We have two equations based on the given data:
1. The area of the ring:
\[ 150 = \pi R^2 - \pi r^2 \]
2. The thickness of the ring:
\[ 3 = R - r \]
Step 4: Solve for the Outer Radius (\( R \))
Rearrange the second equation to express \( r \) in terms of \( R \):
\[ r = R - 3 \]
Substitute this expression for \( r \) in the first equation:
\[ 150 = \pi R^2 - \pi (R - 3)^2 \]
Step 5: Simplify and Solve the Equation
Expand and simplify the equation:
\[ 150 = \pi R^2 - \pi (R^2 - 6R + 9) \]
\[ 150 = \pi R^2 - \pi R^2 + 6\pi R - 9\pi \]
\[ 150 = 6\pi R - 9\pi \]
Divide both sides by \( \pi \):
\[ \frac{150}{\pi} = 6R - 9 \]
\[ \frac{150}{3.14159} = 6R - 9 \]
\[ 47.75 = 6R - 9 \]
Add 9 to both sides:
\[ 47.75 + 9 = 6R \]
\[ 56.75 = 6R \]
Divide by 6:
\[ R = \frac{56.75}{6} \]
\[ R \approx 9.458 \, \text{units} \]
Step 6: Calculate the Inner Radius (\( r \))
Now, use the value of \( R \) to find \( r \):
\[ r = R - 3 \]
\[ r = 9.458 - 3 \]
\[ r \approx 6.458 \, \text{units} \]
Summary
To summarize, the steps to calculate the inner and outer radii of a ring when the area of the ring and the thickness are known are:
1. Use the formulas \(\text{Area of Ring} = \pi R^2 - \pi r^2\) and \(\text{Thickness} = R - r\).
2. Express \( r \) in terms of \( R \).
3. Substitute into the area equation and solve for \( R \).
4. Use the value of \( R \) to find \( r \).
Using our example, with an area of the ring of 150 square units and a thickness of 3 units, we found the outer radius to be approximately 9.458 units and the inner radius to be approximately 6.458 units.
By following these steps, you can easily determine the inner and outer radii of a ring for any given ring area and thickness.
