Volume of Torus Sector given Minor Radius, Lateral Surface Area, and Angle of Intersection Calculator

Formula Used:

\[ V_{Sector} = (2 \times \pi^2 \times \frac{LSA_{Sector}}{4 \times \pi^2 \times r_{Circular\ Section} \times \frac{\angle_{Intersection}}{2\pi}}) \times r_{Circular\ Section}^2 \times \frac{\angle_{Intersection}}{2\pi} \]

Lateral Surface Area of Torus Sector (LSA_Sector):

m²

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

Angle of Intersection of Torus Sector (∠_Intersection):

rad

Volume of Torus Sector (V_Sector):

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1. What is Volume of Torus Sector?

The Volume of Torus Sector refers to the amount of three-dimensional space occupied by a specific sector of a torus. A torus is a doughnut-shaped geometric object, and a sector represents a portion of it defined by an angle of intersection.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

\( V_{Sector} \) — Volume of Torus Sector (m³)
\( LSA_{Sector} \) — Lateral Surface Area of Torus Sector (m²)
\( r_{Circular\ Section} \) — Radius of Circular Section of Torus (m)
\( \angle_{Intersection} \) — Angle of Intersection of Torus Sector (rad)
\( \pi \) — Archimedes' constant (approximately 3.1416)

Explanation: This formula calculates the volume of a torus sector based on its lateral surface area, the radius of its circular cross-section, and the angle of intersection.

3. Importance of Volume Calculation

Details: Calculating the volume of a torus sector is important in various engineering and architectural applications where toroidal structures are used, such as in piping systems, architectural designs, and mechanical components.

4. Using the Calculator

Tips: Enter lateral surface area in m², radius of circular section in m, and angle of intersection 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 of the complete toroidal shape.

Q2: How is this different from a complete torus volume?
A: While a complete torus volume calculates the entire doughnut-shaped object, a torus sector volume calculates only a specific angular portion of it.

Q3: What are typical applications of torus sectors?
A: Torus sectors are commonly used in piping elbows, architectural arches, and various mechanical components that require curved, toroidal sections.

Q4: Can this calculator handle different units?
A: The calculator uses SI units (meters and radians). For other units, convert your measurements to these units before calculation.

Q5: What is the range of valid inputs?
A: All input values must be positive numbers. The angle of intersection should typically be between 0 and 2π radians for meaningful results.