Angle of Intersection of Torus Sector Calculator

Formula Used:

\[ \text{Angle of Intersection} = \frac{LSA_{\text{Sector}}}{4 \times \pi^2 \times r \times r_{\text{Circular Section}}} \times 2\pi \]

Lateral Surface Area of Torus Sector:

m²

Radius of Torus:

Radius of Circular Section of Torus:

Angle of Intersection:

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1. What is the Angle of Intersection of Torus Sector?

The Angle of Intersection of Torus Sector is the angle subtended by the planes in which each of the circular end faces of the Torus Sector is contained. It represents the angular measure between these two intersecting planes.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Angle of Intersection} = \frac{LSA_{\text{Sector}}}{4 \times \pi^2 \times r \times r_{\text{Circular Section}}} \times 2\pi \]

Where:

\( LSA_{\text{Sector}} \) — Lateral Surface Area of Torus Sector (m²)
\( r \) — Radius of Torus (m)
\( r_{\text{Circular Section}} \) — Radius of Circular Section of Torus (m)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: This formula calculates the angle of intersection based on the geometric properties of the torus sector, relating the lateral surface area to the radii of the torus and its circular cross-section.

3. Importance of Angle Calculation

Details: Calculating the angle of intersection is crucial in 3D geometry and engineering applications involving toroidal shapes. It helps in understanding the spatial relationships and designing components with specific angular properties.

4. Using the Calculator

Tips: Enter the lateral surface area of the torus sector in square meters, the radius of the torus in meters, and the radius of the circular section in meters. 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: What units should I use for input values?
A: All input values should be in consistent units (typically meters for lengths and square meters for area). The calculator assumes SI units.

Q3: Can this calculator handle different unit systems?
A: The calculator works with any consistent unit system, but all inputs must use the same units for accurate results.

Q4: What is the range of valid input values?
A: All input values must be positive real numbers greater than zero. The lateral surface area, torus radius, and circular section radius must all be positive.

Q5: How precise are the calculation results?
A: The results are calculated with high precision using PHP's floating-point arithmetic and displayed with 6 decimal places for accuracy.