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How It Works
A circle segment is the region between a chord and the arc it subtends. Three key measurements describe a segment: the chord length (c), the segment height or sagitta (h), and the distance from the center to the chord midpoint (t). These are related to the circle's radius by simple geometric formulas.
The segment area formula uses the inverse cosine (arccos) function to account for the curved boundary. Choose your equation, select which variable to solve for, enter the known values, and click Calculate.
Example Problem
A circle has radius 10 and the perpendicular distance from the center to a chord is 6. Find the chord length and segment height.
- Chord length: c = 2√(10² − 6²) = 2√64 = 16
- Segment height: h = r − t = 10 − 6 = 4
The segment area with r = 10, h = 4 is approximately 44.73 square units.
Frequently Asked Questions
What is the difference between a circle segment and a sector?
A sector is a “pie slice” bounded by two radii and an arc. A segment is the region between a chord and the arc. The segment area equals the sector area minus the triangular area formed by the two radii and the chord.
What is the sagitta of a circle segment?
The sagitta (segment height, h) is the perpendicular distance from the midpoint of the chord to the arc. It equals the radius minus the distance from the center to the chord midpoint: h = r − t.
How do you find the chord length if you know the radius and sagitta?
First calculate t = r − h (the center-to-chord distance). Then apply c = 2√(r² − t²). For r = 10 and h = 3, t = 7, and c = 2√(100 − 49) = 2√51 ≈ 14.28.
Can the segment area formula be solved for radius?
The segment area formula involves both r² and arccos, making it transcendental — there is no closed-form algebraic solution for r given A and h. Numerical methods (Newton's method) are required, so this calculator solves for area given radius and height.
Related Calculators
- Circle Calculator — Calculate area, circumference, radius, and diameter.
- Circle Sector Calculator — Find sector area from radius and central angle.
- Circle Arc Length Calculator — Compute arc length from radius and angle.
- Triangle Calculator — Solve right, equilateral, and scalene triangles.
- Area Converter — Convert between area units.
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