Calculating the area of a ring (also known as an annulus) is a straightforward process if you know the inner and outer radii. Here’s a step-by-step guide to help you through it.
Step 1: Understand the Formula
The area of a ring is found by subtracting the area of the inner circle from the area of the outer circle. The formula is:
\[ \text{Area of Ring} = \pi R^2 - \pi r^2 \]
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 take an example to make this clear. Suppose the outer radius (\( R \)) is 8 units and the inner radius (\( r \)) is 3 units.
Step 3: Calculate the Area of the Outer Circle
First, calculate the area of the outer circle using the formula \( \pi R^2 \):
\[ \text{Area of Outer Circle} = \pi \times R^2 \]
\[ \text{Area of Outer Circle} = \pi \times 8^2 \]
\[ \text{Area of Outer Circle} = \pi \times 64 \]
\[ \text{Area of Outer Circle} = 3.14159 \times 64 \]
\[ \text{Area of Outer Circle} = 201.062 \, \text{square units} \]
Step 4: Calculate the Area of the Inner Circle
Next, calculate the area of the inner circle using the formula \( \pi r^2 \):
\[ \text{Area of Inner Circle} = \pi \times r^2 \]
\[ \text{Area of Inner Circle} = \pi \times 3^2 \]
\[ \text{Area of Inner Circle} = \pi \times 9 \]
\[ \text{Area of Inner Circle} = 3.14159 \times 9 \]
\[ \text{Area of Inner Circle} = 28.274 \, \text{square units} \]
Step 5: Calculate the Area of the Ring
Now, subtract the area of the inner circle from the area of the outer circle to find the area of the ring:
\[ \text{Area of Ring} = \text{Area of Outer Circle} - \text{Area of Inner Circle} \]
\[ \text{Area of Ring} = 201.062 - 28.274 \]
\[ \text{Area of Ring} = 172.788 \, \text{square units} \]
So, the area of the ring is 172.788 square units.
Summary
To summarize, the steps to calculate the area of a ring when the inner and outer radii are known are:
1. Use the formula \( \text{Area of Ring} = \pi R^2 - \pi r^2 \).
2. Calculate the area of the outer circle.
3. Calculate the area of the inner circle.
4. Subtract the area of the inner circle from the area of the outer circle.
Using our example, with an outer radius of 8 units and an inner radius of 3 units, we found the area of the ring to be 172.788 square units.
By following these steps, you can easily calculate the area of a ring for any given inner and outer radii.
