Radius Of Toroid Given Total Surface Area Of Toroid Sector Calculator

Formula Used:

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

Total Surface Area of Toroid Sector:

m²

Cross Sectional Area of Toroid:

m²

Cross Sectional Perimeter of Toroid:

Angle of Intersection of Toroid Sector:

rad

Radius of Toroid:

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1. What is Radius of Toroid?

The radius of a toroid is the distance from the center of the overall toroid to the center of its cross-section. It is a fundamental geometric parameter that defines the size and shape of a toroidal object.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

\( r \) — Radius of Toroid (m)
\( TSA_{Sector} \) — Total Surface Area of Toroid Sector (m²)
\( A_{Cross\ Section} \) — Cross Sectional Area of Toroid (m²)
\( P_{Cross\ Section} \) — Cross Sectional Perimeter of Toroid (m)
\( \angle_{Intersection} \) — Angle of Intersection of Toroid Sector (rad)
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: This formula calculates the radius of a toroid based on the surface area of a toroid sector and its geometric properties.

3. Importance of Toroid Radius Calculation

Details: Calculating the radius of a toroid is essential in various engineering and physics applications, particularly in electromagnetism where toroidal coils are used, and in mechanical design of toroidal components.

4. Using the Calculator

Tips: Enter all values in appropriate units (meters for length, square meters for area, radians for angle). Ensure all values are positive and valid for accurate results.

5. Frequently Asked Questions (FAQ)

Q1: What is a toroid?
A: A toroid is a doughnut-shaped object with a circular cross-section. It's the surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

Q2: What are common applications of toroids?
A: Toroids are commonly used in electromagnetism (toroidal transformers and inductors), mechanical engineering (seals and gaskets), and architectural design.

Q3: How does the angle of intersection affect the calculation?
A: The angle of intersection determines the portion of the toroid being considered. A full toroid would have an angle of 2π radians (360 degrees).

Q4: Can this calculator be used for partial toroids?
A: Yes, this formula specifically calculates the radius for a toroid sector, which can represent a partial toroid.

Q5: What are the limitations of this formula?
A: This formula assumes a perfect toroidal shape with uniform cross-section and may not be accurate for irregular or deformed toroids.