Central Angle Of Annulus Sector Given Diagonal Calculator

Formula Used:

\[ \text{Central Angle of Annulus Sector} = \arccos\left(\frac{r_{\text{Outer}}^2 + r_{\text{Inner}}^2 - d_{\text{Sector}}^2}{2 \times r_{\text{Outer}} \times r_{\text{Inner}}}\right) \]

Outer Circle Radius of Annulus:

Inner Circle Radius of Annulus:

Diagonal of Annulus Sector:

Central Angle of Annulus Sector:

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1. What is the Central Angle of Annulus Sector?

The Central Angle of Annulus Sector is the angle whose vertex is at the center of the concentric circles forming the annulus, and whose sides are radii intersecting the outer and inner circles at distinct points, defining a sector of the annulus.

2. How Does the Calculator Work?

The calculator uses the formula derived from the Law of Cosines:

\[ \theta = \arccos\left(\frac{r_{\text{Outer}}^2 + r_{\text{Inner}}^2 - d_{\text{Sector}}^2}{2 \times r_{\text{Outer}} \times r_{\text{Inner}}}\right) \]

Where:

\( \theta \) — Central Angle of Annulus Sector (radians)
\( r_{\text{Outer}} \) — Outer Circle Radius of Annulus (m)
\( r_{\text{Inner}} \) — Inner Circle Radius of Annulus (m)
\( d_{\text{Sector}} \) — Diagonal of Annulus Sector (m)

Explanation: This formula calculates the central angle by relating the radii of the concentric circles and the diagonal spanning across the annulus sector.

3. Importance of Central Angle Calculation

Details: Calculating the central angle is essential in geometry and engineering for determining sector areas, arc lengths, and in designing components with annular shapes.

4. Using the Calculator

Tips: Enter the outer radius, inner radius, and diagonal length in meters. All values must be positive and satisfy the triangle inequality for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What is an annulus?
A: An annulus is a ring-shaped object, bounded by two concentric circles.

Q2: Can the central angle be greater than 2π radians?
A: In this context, the central angle is typically between 0 and π radians, as it is derived from the cosine inverse function.

Q3: What if the input values do not form a valid triangle?
A: The calculator checks for the triangle inequality and will return an error if the values are not valid.

Q4: How is the diagonal of the annulus sector defined?
A: The diagonal is the straight line distance between two points on the outer and inner arcs at the maximum separation within the sector.

Q5: Can this formula be used for partial annuli?
A: Yes, the formula applies to any sector of an annulus defined by the given parameters.