Radius of Inner Circle of Annulus given Area and Perimeter Calculator

Formula Used:

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

Inner Circle Radius (r_Inner):

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1. What is the Inner Circle Radius of Annulus?

The Inner Circle Radius of Annulus is the radius of its cavity and it is the smaller radius among two concentric circles that form the annulus (ring-shaped region).

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r_{Inner} \) — Inner Circle Radius of Annulus (m)
\( P \) — Perimeter of Annulus (m)
\( A \) — Area of Annulus (m²)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The formula calculates the inner radius by first finding the outer radius from the perimeter, then using the relationship between area, outer radius, and inner radius of the annulus.

3. Importance of Inner Circle Radius Calculation

Details: Calculating the inner circle radius is essential in engineering, architecture, and manufacturing where annular shapes are used. It helps in determining material requirements, structural properties, and design specifications for ring-shaped components.

4. Using the Calculator

Tips: Enter the perimeter and area of the annulus. Both values must be positive numbers. The calculator will compute the inner circle radius using the mathematical relationship between these parameters.

5. Frequently Asked Questions (FAQ)

Q1: What is an annulus?
A: An annulus is a ring-shaped region between two concentric circles with different radii.

Q2: How is the area of annulus related to its radii?
A: Area = π(R² - r²), where R is the outer radius and r is the inner radius.

Q3: What is the perimeter of an annulus?
A: Perimeter = 2π(R + r), the sum of the circumferences of both circles.

Q4: Can this calculator handle different units?
A: The calculator uses consistent units (meters for length, square meters for area). Convert your measurements accordingly before input.

Q5: What if I get a negative result?
A: A negative result indicates invalid input values where the area cannot be contained within the given perimeter for an annulus.