Calculator Features
📐Eight practical solve modes
🧮Step-by-step working
📏Degrees and radians
🌗Minor/major segment comparison
🚫Detects impossible input combinations
🔄Length & area unit conversion
🖨Printable / PDF results
📊CSV export
🖼SVG diagram export
📋Copy results & formula
🎚Decimal precision control
📱Mobile friendly
🧭 Jump to a section
What Is a Circular Segment?
A circular segment is the region between a chord and the arc it cuts off — imagine slicing a circle with a single straight cut; the piece on either side of that cut is a segment. Unlike a sector, a segment's boundary is only the chord and the arc — it does not include the two straight radii. Every segment is fully determined by just two independent measurements: the circle's radius (r) and the central angle (θ) between the chord's endpoints.
A labeled circular segment: radius, chord, arc, sagitta (segment height), central angle, and the shaded segment region itself.
💡 Did You Know? Segment area, unlike an ellipse's perimeter, has a genuine exact closed-form formula — A = ½r²(θ − sinθ) — with no approximation needed. The one exception is solving backward from a known area to the central angle, which this calculator handles numerically and explains plainly, rather than pretending an algebraic shortcut exists.
Segment vs Sector
These are easy to mix up but genuinely different shapes. A sector is the "pie slice" bounded by two radii and the arc between them — it always includes the centre point. A segment is bounded only by a chord and the arc — it never includes the centre unless the segment happens to be a semicircle. A segment's area is always smaller than its sector's area, by exactly the area of the triangle formed by the two radii and the chord.
Segment vs Arc
The arc is a one-dimensional curved line — just a length. The segment is the two-dimensional region enclosed by that arc and its chord — an area. It's a common mix-up: "arc" and "segment" describe fundamentally different kinds of measurement, length versus area, even though they share the same radius and central angle.
Sagitta Explained
The sagitta — also simply called the segment height — is the perpendicular distance from the midpoint of the chord to the midpoint of the arc. It's the same measurement used by the Chord Length Calculator, and it's a genuinely practical alternative starting point: opticians specify lens curvature using sagitta over a known aperture, and engineers describe how far a curved element bulges above a straight baseline the same way.
h = r × (1 − cos(θ/2)) | θ = 2 × arccos((r−h)/r)
Segment Area Formula
A = ½r²(θ − sinθ)
The central angle must be in radians for this formula. A circle with radius 10 and a 60° (≈1.047 rad) central angle has a segment area of ½×100×(1.047−0.866) ≈ 9.059 square units — genuinely different from, and smaller than, that same sector's area of ≈52.360 square units.
Where the Formula Comes From
Segment area is simply sector area minus triangle area. The sector (½r²θ) includes the two straight radii and the pie-slice region between them; subtract the triangle formed by those same two radii and the chord (½r²sinθ), and what's left is exactly the segment. This is genuinely useful to understand, not just memorise — a surprising number of otherwise-thorough calculators state the formula without ever explaining why it works this way.
Segment area = Sector area − Triangle area = ½r²θ − ½r²sinθ = ½r²(θ − sinθ)
Major vs Minor Segment
Any chord divides a circle into two segments: the minor segment (the smaller one, with central angle under 180°) and the major segment (the larger one, over 180°). Both share the exact same chord. This calculator always shows both, since a question that specifies "the segment" without saying which one can be genuinely ambiguous.
Worked Examples
Radius 10, angle 60°
Segment area ≈ 9.059 units², chord = 10, sagitta ≈ 1.340
Partially filled pipe — radius 1, fill depth (sagitta) 0.5
Central angle ≈ 120°, cross-sectional area of the liquid ≈ 0.614 units² — exactly half the pipe's radius filled, giving roughly (but not exactly) half the pipe's area, since fill depth and area aren't linearly related
Lens segment — radius 25, angle 30°
Chord (aperture) ≈ 12.941, sagitta ≈ 0.852, segment area ≈ 7.375 units²
Tunnel cross-section — radius 3, angle 100°
Segment area ≈ 3.422 units² — the flat-bottomed portion cut off a circular bore profile
Arched window — radius 2, angle 140°
Segment area ≈ 3.601 units² — the curved glazing panel above a window's springline
Quarter-circle segment — radius 10, angle 90°
Segment area = ½×100×(π/2−1) ≈ 28.540 units²
Impossible-input example
Radius 10, sagitta entered as 25: a sagitta can never exceed the diameter (20 units for this radius)
The calculator correctly rejects this rather than returning a meaningless angle
Engineering Applications
| Field | Use |
|---|---|
| Structural engineering | Curved beam and arch cross-sections, load-bearing curve geometry |
| Fluid engineering | Partially filled horizontal pipe or tank cross-sectional area, directly a segment calculation |
| Optics | Lens and mirror curvature specified by sagitta over a known aperture chord |
| Manufacturing | Sheet-metal segment blanks, curved panel cutting, CNC-machined curved pockets |
| Architecture | Arched window glazing above the springline, dome cross-sections, fan windows |
| CAD design | Segment entities defined directly by radius, chord, or sagitta, depending on the design system |
Surveying Applications
| Use | How segment geometry helps |
|---|---|
| Road and tunnel cross-sections | Flat-bottomed bored tunnels and road drainage channels are often calculated as a full circle minus a segment |
| Curved boundary surveys | Land area enclosed by a curved boundary and a straight chord line uses segment area directly |
| Large-radius estimates | For gentle curves (large radius, small angle), segment area is a small correction on top of the triangle area — useful for quick field checks |
Glossary
- Circular segment
- The region between a chord and its arc, not including the two radii.
- Circular sector
- The "pie slice" region between two radii and the arc connecting them, including the straight radii.
- Chord
- The straight line segment connecting the two endpoints of the arc.
- Sagitta
- The perpendicular height between the midpoint of the chord and the midpoint of the arc; also called segment height.
- Central angle
- The angle at the circle's centre between the two radii drawn to the chord's endpoints.
- Minor segment
- The smaller of the two segments formed by a chord (central angle under 180°).
- Major segment
- The larger of the two segments formed by the same chord (central angle over 180°).
- Distance from centre
- The perpendicular distance from the circle's centre to the chord line.
- Triangle area (of a segment)
- The area of the isosceles triangle formed by the two radii and the chord — the piece subtracted from the sector to leave the segment.
Common Mistakes
| Mistake | Fix |
|---|---|
| Confusing segment area with sector area | A sector includes the two radii; a segment doesn't — the segment is always the smaller of the two for the same angle |
| Using degrees directly in the segment area formula | A = ½r²(θ−sinθ) requires θ in radians — convert first by multiplying degrees by π/180 |
| Assuming segment area scales linearly with sagitta | The relationship between height and area is not proportional — doubling the sagitta does not double the area |
| Forgetting there are two segments for any chord | The minor and major segments share the same chord but have very different areas |
| Mixing up segment perimeter with just the arc length | Segment perimeter includes both the arc and the straight chord: P = arc length + chord |
Formula Cheat Sheet
Quick Reference
Segment area:A = ½r²(θ−sinθ) | Sector area: A = ½r²θ | Triangle area: A = ½r²|sinθ|Chord:
c = 2r×sin(θ/2) | Sagitta: h = r(1−cos(θ/2)) | Arc length: L = rθSegment perimeter:
P = L + c | Angle from sagitta: θ = 2×arccos((r−h)/r) | Radius from chord+sagitta: r = c²/(8h)+h/2
Practice Questions
Beginner (with answers)
- Find the segment area for radius 8 and central angle 90°.
- Find the segment area for radius 6 and central angle 60°.
- Is segment area ever larger than sector area for the same angle?
- Find the sagitta for radius 10, central angle 90°.
- What is the segment area for a semicircle of radius 5?
Show answers
1) ½×64×(π/2−1)≈18.265 2) ½×36×(π/3−sin60°)≈3.261 3) No — segment is always smaller for the same angle, under 360° 4) 10×(1−cos45°)≈2.929 5) πr²/2=π×25/2≈39.270
Advanced (with answers)
- A segment has radius 10 and area 9.06. Find the central angle in degrees.
- A segment has chord 16 and sagitta 4. Find the radius.
- Find the triangle area for radius 10, central angle 300°.
- A segment has radius 12, distance from centre 6. Find the chord.
- Find the segment perimeter for radius 10, central angle 60°.
Show answers
1) Solved numerically: θ≈60° (matches the reference case) 2) r=16²/(8×4)+4/2=8+2=10 3) ½×100×|sin300°|≈43.301 (same physical triangle as the 60° case) 4) c=2√(144−36)≈20.785 5) arc+chord=10.472+10=20.472
🔑 Key Takeaways
- Segment area A = ½r²(θ−sinθ) is exact — no approximation needed for any forward calculation.
- A segment excludes the two radii that a sector includes; segment area = sector area − triangle area.
- Sagitta (segment height) is a genuinely practical alternative starting point, widely used in optics and engineering.
- The same chord always produces two segments — minor and major — sharing one chord but very different areas.
- Solving backward from a known area to the central angle has no algebraic shortcut — this calculator solves that combination numerically and says so.
Frequently Asked Questions
The region between a chord and the arc it cuts off — imagine slicing a circle with a single straight cut; the piece on either side of that cut is a segment.
A = ½r²(θ − sinθ), using the radius and central angle in radians.
A = r²×arccos((r−h)/r) − (r−h)×√(2rh−h²), using the radius and segment height (sagitta) directly.
A sector includes the two straight radii as part of its boundary; a segment is bounded only by the chord and the arc, with no radii.
The larger of the two segments formed by a chord, corresponding to a central angle greater than 180°.
The smaller of the two segments formed by a chord, corresponding to a central angle less than 180°.
h = r − √(r² − (c/2)²), using the radius and chord length — this is also called the sagitta.
θ = 2×arcsin(c/2r), which gives the minor angle; the major angle is 360° minus that result.
Because segment area equals sector area minus the triangle formed by the two radii and the chord, and that triangle's area is ½r²sinθ.
Yes for any angle under 360° — the segment excludes the triangular portion the sector includes.
The perpendicular height between the midpoint of a chord and the midpoint of its arc — the same measurement as segment height.
To calculate cross-sectional areas of curved structural elements, partially filled pipes, and tank liquid volumes.
For arched window glazing above the springline, curved facade panels, and dome cross-sections.
Curved property boundaries and road-curve offsets are calculated using segment and chord geometry together.
Sheet-metal segment blanks and CNC-cut curved panels are dimensioned using segment area and chord or height measurements.
No — the diameter is the longest possible chord in any circle; entering a longer value is geometrically impossible.
Yes, for a major segment (central angle over 180°) — but it can never exceed the full diameter.
Just under the full circle's area (πr²), approached as the central angle approaches 360°.
Any squared linear unit — mm², cm², m², km², or the imperial equivalents in², ft², yd².
The core formula A = ½r²(θ−sinθ) requires radians; convert degrees first by multiplying by π/180.
A sector minus its enclosed isosceles triangle (formed by the two radii and the chord) equals the segment.
Both, simultaneously — a semicircle is the special case where segment and sector coincide, since the chord (the diameter) passes through the centre.
Add the arc length to the chord length: P = arc length + chord.
An arc is a curved line — one-dimensional, a length; a segment is the enclosed area bounded by that arc and its chord.
This requires solving the transcendental equation numerically for r, since it cannot be rearranged algebraically the way chord+sagitta can.
There's no fixed minimum — segment area approaches zero as the central angle approaches zero.
No — a valid central angle must be greater than 0° and no more than 360°.
The "segment" becomes the entire circle, and the chord degenerates to a single point.
A properly built one uses exact closed-form trigonometric formulas for every forward calculation, accurate to floating-point precision — no approximation needed for the core cases.
They're the same measurement — "sagitta" is the traditional geometric term, and "segment height" is the more common everyday name.
The cross-sectional area of liquid in a horizontally-lying pipe, filled to a certain depth, is exactly a circular segment area calculation.
Lens and mirror curvature is often specified using the sagitta over a known aperture (chord) width.
Yes — the same formulas apply directly when the central angle exceeds 180°, without needing a separate formula.
A chord is the line segment between two points on a circle; a secant is the full infinite line extended through those same two points.
Multiply by the square of the linear conversion factor, never the plain linear factor — converting length units directly for an area value gives a wrong result.
Yes — it's a standard topic in circle geometry units, typically alongside sectors and arc length.
Segment area equals the sector's ½r²θ minus the enclosed triangle's ½r²sinθ, combining to ½r²(θ−sinθ).
Yes — the radius itself can be derived first: r = c²/(8h) + h/2, then every other property follows.
The minor arc is just the curved boundary line; the minor segment is the full enclosed area bounded by that arc and the chord.
Curved bridge deck cross-sections and arch rib profiles are described and quantified using segment geometry.
Tunnel cross-sections that aren't full circles, such as flat-bottomed bored tunnels, are often calculated as a circle minus a segment.
Road cross-fall and curved kerb cross-sections sometimes use segment geometry for drainage capacity calculations.
Curved pocket or slot cross-sections cut by CNC tooling are dimensioned and verified using segment area formulas.
No — any two independent, compatible measurements (radius+angle, radius+height, chord+height, and more) are enough to solve the whole shape.
A = ½r²(θ−sinθ) is exact for every forward calculation — unlike an ellipse's perimeter, a circular segment's area has a genuine closed-form solution with no approximation caveat needed.
A = ½r²(π/2 − 1) — for r=10, that's approximately 28.540 square units.
Yes — a one-page reference covering every segment formula, with a QR code linking back to the live calculator, is available further down this page.
Yes — copy to clipboard, CSV export, SVG diagram export, and a printable PDF-ready worksheet are all available on this calculator.
Exactly half the circle's area, πr²/2, since a 180° "segment" is the full half-circle with the chord (diameter) contributing zero triangle area.
A segment is a single-circle, chord-cut region; an annulus is the ring-shaped region between two different concentric circles — entirely different shapes despite both relating to circles.
The calculator uses the required values for the selected mode to solve the segment, and cross-checks any extra values you entered (chord or perimeter) against the result, flagging a warning if they don't match.
A segment only has two independent measurements (radius and central angle, or any equivalent pair), so any two compatible values determine it fully. Eight combinations are offered because segment problems arise from very different starting values in practice — sometimes an angle, sometimes a sagitta, sometimes a known area that needs solving backward.
Calculation Assumptions
ℹ️ What this calculator assumes
- Results are calculated using Euclidean (flat-plane) geometry.
- Measurements are assumed to lie in a flat plane, not on a curved or sloped surface.
- Rounding to the selected decimal precision may cause very small differences between displayed values and hand calculations.
- All input values are assumed to use the same unit — mixing units (e.g. a radius in metres and a chord in centimetres) will produce an incorrect result.
- Seven of the eight solve modes have exact closed-form solutions. Radius + Segment Area is solved by numerical bisection, since no algebraic inverse exists for θ − sinθ = 2A/r².
- The Chord + Radius mode always returns the minor central angle; the major (reflex) angle and its segment are shown separately in the comparison panel.
About This Calculator
Printable Formula Sheet
A one-page reference with every circular segment formula on this page.
Circular Segment Formula Sheet
MegaCalcOnline.com · Segment area, sagitta, chord and perimeter formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Segment area | A = ½r²(θ−sinθ) |
| Sector area | A = ½r²θ |
| Triangle area | A = ½r²|sinθ| |
| Chord length | c = 2r×sin(θ/2) |
| Sagitta (segment height) | h = r(1−cos(θ/2)) |
| Arc length | L = rθ |
| Segment perimeter | P = L + c |
| Angle from sagitta | θ = 2×arccos((r−h)/r) |
| Angle from chord + radius | θ = 2×arcsin(c/2r) |
| Radius from chord + sagitta | r = c²/(8h) + h/2 |
References
- Khan Academy — Circle segments and sectors
- OpenStax Mathematics — circular measurement
- Wolfram MathWorld — Circular Segment
- NIST — SI units for length and area
- Maths Is Fun — Circle Sector and Segment
- Engineers Edge — Circular Segment Equation and Calculator
Every formula and worked example on this page was independently verified — see “About This Calculator” above for the full review notes.