Radius of Circular Section of Torus given Volume of Torus Sector Calculator

Radius of Circular Section Formula:

\[ r = \sqrt{\frac{V}{2\pi^2 R \left(\frac{\theta}{2\pi}\right)}} \]

Volume of Torus Sector:

m³

Radius of Torus:

Angle of Intersection:

rad

Tolerance:

Radius of Circular Section (m):

1. What is Radius of Circular Section of Torus?

Definition: The radius of the circular cross-section of a torus (donut shape) calculated from the volume of a torus sector.

Purpose: This calculation is important in geometry and engineering when working with torus-shaped objects where only a sector's volume is known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r = \sqrt{\frac{V}{2\pi^2 R \left(\frac{\theta}{2\pi}\right)}} \]

Where:

\( r \) — Radius of circular section (meters)
\( V \) — Volume of torus sector (cubic meters)
\( R \) — Radius of torus (meters)
\( \theta \) — Angle of intersection (radians)

Explanation: The formula derives the cross-section radius from the sector volume, torus radius, and angle of the sector.

3. Importance of Tolerance

Details: The ±5% tolerance (default) accounts for manufacturing variations and measurement uncertainties in practical applications.

4. Using the Calculator

Tips: Enter the volume of torus sector, torus radius, angle of intersection (in radians), and tolerance percentage. All values must be > 0.

5. Frequently Asked Questions (FAQ)

Q1: What is a torus sector?
A: A torus sector is a portion of a torus bounded by two planes intersecting the torus at a specific angle.

Q2: Why is the angle in radians?
A: Radians are the natural unit for angular measurement in mathematical calculations involving circles and circular sections.

Q3: How do I convert degrees to radians?
A: Multiply degrees by π/180 (approximately 0.0174533).

Q4: When would I adjust the tolerance?
A: Increase tolerance for rough estimates or decrease for precision applications.

Q5: Can this calculator handle very small or large values?
A: Yes, but extremely large values may exceed floating-point precision limits.