Calculator Inputs
Choose one solving method. The form shows only the inputs needed for that method.
Formula Used
Different input combinations solve the same geometric quantity. The calculator also derives the related chord, area, arc length, and angle.
Primary height formulas
Minor segment height from radius and chord: h = r - √(r² - (c/2)²)
Height from true central angle: h = r(1 - cos(θ/2))
Height from arc length: h = r(1 - cos((s/r)/2))
Segment area relation: A = (r² / 2)(θ - sin θ)
Derived geometry
Chord from radius and height: c = 2√(2rh - h²)
Central angle from radius and height: θ = 2 arccos(1 - h/r)
Arc length: s = rθ
Segment perimeter: P = c + s
For area input, the calculator solves the area equation numerically because the angle cannot be isolated with a simple elementary formula.
How to Use This Calculator
- Choose the method matching the measurements you already know.
- Enter a unit label so the output reads clearly.
- Set the number of decimal places for final rounding.
- For chord methods, pick minor or major segment mode.
- Enter the known values and submit the form.
- Review the result block above the form for the height.
- Use the data table, graph, and derived values for checking.
- Download the result table as CSV or PDF when needed.
Example Data Table
| Method | Known Inputs | Segment Type | Computed Height | Notes |
|---|---|---|---|---|
| Radius + Chord | r = 10, c = 12 | Minor | 2 | Common sagitta case for fabrication work. |
| Radius + Central Angle | r = 8, θ = 90° | Minor | 2.3431 | Direct trigonometric solution. |
| Diameter + Chord | d = 20, c = 16 | Minor | 4 | Diameter first becomes radius. |
| Radius + Arc Length | r = 12, s = 10 | Minor | 1.0269 | Arc converts to angle using s / r. |
FAQs
1) What is a circle segment height?
Circle segment height is the perpendicular distance from a chord to the arc directly above it. Many engineers also call it the sagitta for the minor segment.
2) When should I choose minor or major segment mode?
Use minor mode when the curved portion is smaller than a semicircle. Use major mode when the segment wraps around more than half the circle and the height exceeds the radius.
3) Can the segment height be larger than the radius?
Yes. That happens for a major segment. Minor segments always have heights from zero up to the radius, while major segments range from the radius up to almost twice the radius.
4) Why does the area method use numerical solving?
The segment area equation contains both the angle and its sine. That combination does not rearrange into a simple direct inverse, so iterative solving gives the practical answer.
5) Which units should I enter?
You can use any consistent length unit, such as millimeters, centimeters, meters, or inches. Area results automatically appear in squared versions of the same chosen unit label.
6) Does the central angle accept degrees and radians?
Yes. Pick the angle unit before submitting. The calculator converts everything internally, then reports the final angle in both degrees and radians for easier checking.
7) What happens when the chord equals the diameter?
That creates a semicircle. The segment height becomes exactly equal to the radius, and the central angle becomes 180 degrees.
8) Is this calculator useful for design and machining checks?
Yes. It is useful for drafting, machining, fabrication, arch layouts, tank profiles, and geometric verification. Still, always apply your project’s required tolerance and rounding policy.