Central Angle Of Annulus Sector Given Diagonal And Outer Circle Radius Calculator

Formula Used:

\[ \text{Central Angle of Annulus Sector} = \cos^{-1}\left(1-\frac{\text{Diagonal of Annulus Sector}^2-\text{Breadth of Annulus}^2}{2\times\text{Outer Circle Radius of Annulus}\times(\text{Outer Circle Radius of Annulus}-\text{Breadth of Annulus})}\right) \]

Central Angle of Annulus Sector:

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1. What is Central Angle of Annulus Sector?

The Central Angle of Annulus Sector is the angle whose apex (vertex) is the center of the concentric circles of Annulus and whose legs (sides) are radii intersecting the circles in four distinct points.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \angle_{\text{Central(Sector)}} = \cos^{-1}\left(1-\frac{d_{\text{Sector}}^2-b^2}{2\times r_{\text{Outer}}\times(r_{\text{Outer}}-b)}\right) \]

Where:

\( d_{\text{Sector}} \) — Diagonal of Annulus Sector (m)
\( b \) — Breadth of Annulus (m)
\( r_{\text{Outer}} \) — Outer Circle Radius of Annulus (m)

Explanation: The formula calculates the central angle of an annulus sector using the diagonal length, breadth of the annulus, and the outer circle radius through inverse cosine trigonometric function.

3. Importance of Central Angle Calculation

Details: Calculating the central angle is crucial for determining the area and perimeter of annulus sectors, which has applications in engineering, architecture, and various geometric calculations involving circular segments.

4. Using the Calculator

Tips: Enter diagonal of annulus sector and outer circle radius in meters (must be positive values). The breadth of annulus must be less than the outer circle radius. All values must be valid for accurate calculation.

5. Frequently Asked Questions (FAQ)

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

Q2: What units should I use for input values?
A: The calculator uses meters as the default unit, but you can use any consistent unit system as long as all measurements are in the same units.

Q3: Can the breadth be zero?
A: The breadth can be zero theoretically (when both circles have the same radius), but in practice, this would represent a line rather than an annulus.

Q4: What is the range of possible central angle values?
A: The central angle can range from 0° to 360°, representing the full circular sector of the annulus.

Q5: Are there any limitations to this formula?
A: The formula assumes perfect concentric circles and may not be accurate for non-circular or eccentric annulus shapes.