Radius of Inner Circle of Annulus given Longest Interval and Perimeter Calculator

Formula Used:

\[ r_{Inner} = \frac{\left(\frac{P}{2\pi}\right) - \left(\frac{\frac{l^2}{4}}{\frac{P}{2\pi}}\right)}{2} \]

Inner Circle Radius of Annulus (r_Inner):

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1. What is the Annulus Inner Circle Radius Formula?

The formula calculates the inner circle radius of an annulus given its perimeter and the longest interval (chord tangent to the inner circle). An annulus is the region between two concentric circles.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_{Inner} = \frac{\left(\frac{P}{2\pi}\right) - \left(\frac{\frac{l^2}{4}}{\frac{P}{2\pi}}\right)}{2} \]

Where:

\( r_{Inner} \) — Inner circle radius of annulus (m)
\( P \) — Perimeter of annulus (m)
\( l \) — Longest interval of annulus (m)
\( \pi \) — Mathematical constant pi (≈3.14159)

Explanation: The formula derives from geometric relationships between the perimeter, longest chord, and radii of concentric circles.

3. Importance of Inner Circle Radius Calculation

Details: Calculating the inner radius is essential for determining the area of the annulus, understanding geometric properties, and solving practical engineering problems involving circular rings.

4. Using the Calculator

Tips: Enter the perimeter and longest interval values in meters. Both values must be positive numbers. The calculator will compute the inner circle radius.

5. Frequently Asked Questions (FAQ)

Q1: What is an annulus?
A: An annulus is the region between two concentric circles, resembling a ring or washer shape.

Q2: How is the longest interval defined?
A: The longest interval is the chord tangent to the inner circle, which is the longest possible straight line segment that can be drawn within the annulus.

Q3: Can this formula be used for any annulus?
A: Yes, this formula applies to all concentric circular annuli where the perimeter and longest interval are known.

Q4: What are practical applications of this calculation?
A: This calculation is used in engineering, architecture, and manufacturing where annular shapes are common, such as in pipe systems, bearings, and mechanical seals.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the input values, though practical accuracy depends on the precision of the measurements provided.