Determining the outer radius of a ring (annulus) is straightforward if you know the inner area and the ring area. Follow this step-by-step guide to understand how to do it.
Step 1: Understand the Formula
To find the outer radius, you need to use the given inner area and ring area. The formulas you will use are:
\[ \text{Inner Area} = \pi r^2 \]
\[ \text{Ring Area} = \pi R^2 - \text{Inner Area} \]
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 inner area is 28.274 square units and the ring area is 113.097 square units.
Step 3: Calculate the Inner Radius
First, determine the inner radius (\( r \)) from the given inner area:
\[ \text{Inner Area} = \pi r^2 \]
\[ 28.274 = 3.14159 \times r^2 \]
\[ r^2 = \frac{28.274}{3.14159} \]
\[ r^2 = 9 \]
\[ r = \sqrt{9} \]
\[ r = 3 \, \text{units} \]
Step 4: Calculate the Area of the Outer Circle
Next, use the ring area and the inner area to calculate the area of the outer circle:
\[ \text{Ring Area} = \pi R^2 - \text{Inner Area} \]
\[ 113.097 = \pi R^2 - 28.274 \]
\[ \pi R^2 = 113.097 + 28.274 \]
\[ \pi R^2 = 141.371 \]
Step 5: Calculate the Outer Radius
Now, solve for the outer radius (\( R \)):
\[ R^2 = \frac{141.371}{3.14159} \]
\[ R^2 = 45 \]
\[ R = \sqrt{45} \]
\[ R = 6.708 \, \text{units} \]
So, the outer radius of the ring is approximately 6.708 units.
Summary
To summarize, the steps to calculate the outer radius of a ring when the inner area and ring area are known are:
1. Use the formula \( \text{Inner Area} = \pi r^2 \) to find the inner radius.
2. Calculate the area of the outer circle using the ring area and the inner area.
3. Solve for the outer radius.
Using our example, with an inner area of 28.274 square units and a ring area of 113.097 square units, we found the outer radius to be approximately 6.708 units.
By following these steps, you can easily determine the outer radius of a ring for any given inner area and ring area.
