Calculating a Segment of Circle

Segment of Circle Formula:

\[ A = r^2 \cos^{-1}\left(\frac{r - h}{r}\right) - (r - h) \sqrt{2 r h - h^2} \]

Segment Area (A):

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1. What is a Circular Segment?

A circular segment is the region of a circle bounded by a chord and the arc subtended by that chord. The sagitta (h) is the height of the segment, measured from the chord to the arc.

2. How Does the Calculator Work?

The calculator uses the segment area formula:

\[ A = r^2 \cos^{-1}\left(\frac{r - h}{r}\right) - (r - h) \sqrt{2 r h - h^2} \]

Where:

\( A \) — Area of the segment
\( r \) — Radius of the circle
\( h \) — Sagitta (height of the segment)

Explanation: The formula combines inverse trigonometric and algebraic functions to calculate the area between the chord and the arc.

3. Applications of Circular Segments

Details: Circular segments are used in engineering (arch design), architecture (window and door designs), and physics (lens calculations).

4. Using the Calculator

Tips: Enter the radius and sagitta in consistent length units. The sagitta must be between 0 and 2×radius.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between a segment and a sector?
A: A sector includes two radii and the arc, while a segment is bounded by a chord and the arc.

Q2: What if my sagitta is greater than the radius?
A: The formula works for all sagitta values up to the diameter (2×radius).

Q3: How is this related to the area of a circular cap?
A: A circular cap is a special case of a segment where the chord is the diameter.

Q4: Can I calculate the chord length from this?
A: Yes, the chord length can be derived as \( c = 2\sqrt{2 r h - h^2} \).

Q5: What units should I use?
A: Any consistent length units can be used (meters, feet, etc.), but all inputs must be in the same units.