Radius of Torus given Volume of Torus Sector Calculator

Formula Used:

\[ r = \frac{V_{Sector}}{2 \times \pi^2 \times r_{Circular\ Section}^2 \times \frac{\angle_{Intersection}}{2\pi}} \]

Volume of Torus Sector (V_Sector):

m³

Radius of Circular Section of Torus (r_{Circular Section}):

Angle of Intersection of Torus Sector (∠_Intersection):

rad

Radius of Torus (r):

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1. What is the Radius of Torus given Volume of Torus Sector Formula?

The formula calculates the radius of a torus when given the volume of a torus sector, the radius of the circular cross-section, and the angle of intersection. This is useful in geometric calculations involving toroidal shapes.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r = \frac{V_{Sector}}{2 \times \pi^2 \times r_{Circular\ Section}^2 \times \frac{\angle_{Intersection}}{2\pi}} \]

Where:

\( r \) — Radius of Torus (m)
\( V_{Sector} \) — Volume of Torus Sector (m³)
\( r_{Circular\ Section} \) — Radius of Circular Section of Torus (m)
\( \angle_{Intersection} \) — Angle of Intersection of Torus Sector (rad)
\( \pi \) — Archimedes' constant (≈3.14159)

Explanation: The formula derives the main torus radius from the volume of a sector by accounting for the circular cross-section dimensions and the angular portion of the torus.

3. Importance of Torus Radius Calculation

Details: Calculating the torus radius is essential in various engineering and mathematical applications, particularly in designing toroidal structures and analyzing their geometric properties.

4. Using the Calculator

Tips: Enter volume in cubic meters, circular section radius in meters, and angle in radians. All values must be positive numbers.

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 that intersect the torus, creating a specific angular section.

Q2: Why is the angle measured in radians?
A: Radians are used because they provide a natural measurement of angle in mathematical formulas involving circles and circular motion.

Q3: Can this formula be used for partial torus volumes?
A: Yes, this formula specifically calculates the main radius when given the volume of a torus sector (partial volume).

Q4: What are typical applications of this calculation?
A: This calculation is used in mechanical engineering, architecture, and physics where toroidal shapes are encountered.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, assuming precise input values and using the exact value of π in computations.