Calculating the area of a ring (also known as an annulus) is straightforward if you know the outer area and the inner radius. Here’s a step-by-step guide to help you through the process.
Step 1: Understand the Formula
To determine the area of the ring, you first need to find the outer radius using the given outer area. The formulas involved are:
\[ \text{Outer Area} = \pi R^2 \]
where:
- \( R \) is the outer radius
- \( \pi \) (Pi) is approximately 3.14159
Then, use the outer radius to find the area of the ring:
\[ \text{Area of Ring} = \pi R^2 - \pi r^2 \]
where:
- \( r \) is the inner radius
Step 2: Use Real Numbers for Calculation
Let's take an example to clarify. Suppose the outer area is 314.159 square units and the inner radius (\( r \)) is 6 units.
Step 3: Calculate the Outer Radius
First, use the outer area to calculate the outer radius (\( R \)):
\[ \text{Outer Area} = \pi R^2 \]
\[ 314.159 = 3.14159 \times R^2 \]
\[ R^2 = \frac{314.159}{3.14159} \]
\[ R^2 = 100 \]
\[ R = \sqrt{100} \]
\[ R = 10 \, \text{units} \]
Step 4: Calculate the Area of the Inner Circle
Next, calculate the area of the inner circle using the formula \( \pi r^2 \):
\[ \text{Area of Inner Circle} = \pi \times r^2 \]
\[ \text{Area of Inner Circle} = \pi \times 6^2 \]
\[ \text{Area of Inner Circle} = \pi \times 36 \]
\[ \text{Area of Inner Circle} = 3.14159 \times 36 \]
\[ \text{Area of Inner Circle} = 113.097 \, \text{square units} \]
Step 5: Calculate the Area of the Ring
Now, subtract the area of the inner circle from the outer area to find the area of the ring:
\[ \text{Area of Ring} = \text{Outer Area} - \text{Area of Inner Circle} \]
\[ \text{Area of Ring} = 314.159 - 113.097 \]
\[ \text{Area of Ring} = 201.062 \, \text{square units} \]
So, the area of the ring is 201.062 square units.
Summary
To summarize, the steps to calculate the area of a ring when the outer area and inner radius are known are:
1. Use the formula \( \text{Outer Area} = \pi R^2 \) to find the outer radius.
2. Calculate the area of the inner circle.
3. Subtract the area of the inner circle from the outer area.
Using our example, with an outer area of 314.159 square units and an inner radius of 6 units, we found the area of the ring to be 201.062 square units.
By following these steps, you can easily calculate the area of a ring for any given outer area and inner radius.
