Diagonal of Annulus Sector Calculator

Diagonal of Annulus Sector Formula:

\[ d_{Sector} = \sqrt{r_{Outer}^2 + r_{Inner}^2 - 2 \times r_{Outer} \times r_{Inner} \times \cos(\angle_{Central(Sector)})} \]

Outer Circle Radius of Annulus:

Inner Circle Radius of Annulus:

Central Angle of Annulus Sector:

rad

Diagonal of Annulus Sector:

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1. What is the Diagonal of Annulus Sector?

The Diagonal of Annulus Sector is a line segment joining the two opposite points, at the maximum distance, on the outer and inner arc of an annulus sector. It represents the longest chord that can be drawn within the sector of an annulus.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ d_{Sector} = \sqrt{r_{Outer}^2 + r_{Inner}^2 - 2 \times r_{Outer} \times r_{Inner} \times \cos(\angle_{Central(Sector)})} \]

Where:

\( d_{Sector} \) — Diagonal of Annulus Sector (m)
\( r_{Outer} \) — Outer Circle Radius of Annulus (m)
\( r_{Inner} \) — Inner Circle Radius of Annulus (m)
\( \angle_{Central(Sector)} \) — Central Angle of Annulus Sector (rad)

Explanation: This formula is derived from the Law of Cosines, applied to the triangle formed by connecting the endpoints of the diagonal with the center of the annulus.

3. Importance of Diagonal Calculation

Details: Calculating the diagonal of an annulus sector is important in various engineering and architectural applications, particularly in designing curved structures, mechanical components, and geometric analysis of annular sections.

4. Using the Calculator

Tips: Enter outer radius and inner radius in meters, central angle in radians. All values must be positive (radii > 0, angle ≥ 0). The central angle should be between 0 and 2π radians.

5. Frequently Asked Questions (FAQ)

Q1: What is an annulus sector?
A: An annulus sector is the portion of an annulus (ring-shaped object) bounded by two radii and the corresponding arcs.

Q2: Can I use degrees instead of radians?
A: The calculator requires radians. To convert degrees to radians, multiply by π/180.

Q3: What is the range of valid values for the central angle?
A: The central angle should be between 0 and 2π radians (0° to 360°).

Q4: What if the inner radius is larger than the outer radius?
A: The calculator requires outer radius > inner radius > 0 for a valid annulus.

Q5: How accurate is the calculation?
A: The calculation uses precise trigonometric functions and provides results accurate to 6 decimal places.