Volume Of Torus Sector Given Lateral Surface Area And Radius Calculator

Volume of Torus Sector Formula:

\[ V_{Sector} = 2\pi^2 r \left( \frac{LSA_{Sector}}{4\pi^2 r \left( \frac{\angle_{Intersection}}{2\pi} \right)} \right)^2 \left( \frac{\angle_{Intersection}}{2\pi} \right) \]

Radius of Torus (r):

Lateral Surface Area of Torus Sector (LSA_Sector):

m²

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 represents the amount of three-dimensional space occupied by a specific sector of a torus. A torus is a doughnut-shaped surface generated by rotating a circle in three-dimensional space about an axis coplanar with the circle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ V_{Sector} = 2\pi^2 r \left( \frac{LSA_{Sector}}{4\pi^2 r \left( \frac{\angle_{Intersection}}{2\pi} \right)} \right)^2 \left( \frac{\angle_{Intersection}}{2\pi} \right) \]

Where:

\( V_{Sector} \) — Volume of Torus Sector (m³)
\( r \) — Radius of Torus (m)
\( LSA_{Sector} \) — Lateral Surface Area of Torus Sector (m²)
\( \angle_{Intersection} \) — Angle of Intersection of Torus Sector (rad)
\( \pi \) — Archimedes' constant (≈ 3.14159)

Explanation: The formula calculates the volume of a torus sector based on the given lateral surface area, torus radius, and intersection angle.

3. Importance of Volume Calculation

Details: Calculating the volume of torus sectors is important in various engineering and architectural applications, particularly in the design of curved structures, piping systems, and mechanical components with toroidal shapes.

4. Using the Calculator

Tips: Enter the radius of torus in meters, lateral surface area in square meters, 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 the complete torus volume is calculated using \( V = 2\pi^2 R r^2 \), the sector volume represents only a fractional part of the complete volume based on the intersection angle.

Q3: What are typical applications of torus sector calculations?
A: These calculations are used in architectural design, mechanical engineering (particularly for curved piping and ducting), and in various manufacturing processes involving toroidal components.

Q4: Can this calculator handle different units?
A: The calculator uses meters for length units. For other units, convert your measurements to meters before inputting them into the calculator.

Q5: What if I have the angle in degrees instead of radians?
A: Convert degrees to radians by multiplying by π/180 before entering the angle value into the calculator.