Radius Of Toroid Given Volume Of Toroid Sector Calculator

Formula Used:

\[ r = \frac{V_{Sector}}{2\pi \times A_{Cross\ Section} \times \left(\frac{\angle_{Intersection}}{2\pi}\right)} \]

Volume of Toroid Sector (V_Sector):

m³

Cross Sectional Area of Toroid (A_{Cross Section}):

m²

Angle of Intersection of Toroid Sector (∠_Intersection):

rad

Radius of Toroid (r):

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1. What is the Radius of Toroid Formula?

The formula calculates the radius of a toroid given the volume of a toroid sector, cross-sectional area, and the angle of intersection. This is essential in geometry and engineering applications involving toroidal shapes.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r = \frac{V_{Sector}}{2\pi \times A_{Cross\ Section} \times \left(\frac{\angle_{Intersection}}{2\pi}\right)} \]

Where:

\( r \) — Radius of Toroid (m)
\( V_{Sector} \) — Volume of Toroid Sector (m³)
\( A_{Cross\ Section} \) — Cross Sectional Area of Toroid (m²)
\( \angle_{Intersection} \) — Angle of Intersection of Toroid Sector (rad)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: The formula derives from the geometric properties of toroids and their sectors, relating the overall radius to the volume and cross-sectional parameters.

3. Importance of Radius Calculation

Details: Accurate radius calculation is crucial for designing toroidal components in various engineering fields, including electromagnetics, mechanical design, and architectural structures.

4. Using the Calculator

Tips: Enter volume in cubic meters, cross-sectional area in square meters, and angle in radians. All values must be positive and valid for accurate results.

5. Frequently Asked Questions (FAQ)

Q1: What is a toroid?
A: A toroid is a doughnut-shaped solid generated by revolving a circle around an axis external to the circle.

Q2: What units should I use for inputs?
A: Use consistent SI units: cubic meters for volume, square meters for area, and radians for angle.

Q3: Can I use degrees instead of radians?
A: The calculator requires radians. Convert degrees to radians by multiplying by π/180.

Q4: What if I get a negative result?
A: All input values must be positive. Negative results indicate invalid input data.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact based on the formula, assuming precise input values.