Calculator Features
📐Eight practical solve modes
🧮Step-by-step working
📏Degrees and radians
🌗Minor/major arc 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
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What Is a Chord?
A chord is a straight line segment connecting any two points on a circle's circumference. Unlike a radius or diameter, a chord doesn't need to pass through the centre — though when it does, it becomes the special case of a diameter. Every chord is fully determined by just two independent measurements: the circle's radius (r) and the central angle (θ) between the two points it connects.
💡 Did You Know? The same chord always corresponds to two different arcs — a shorter minor arc and a longer major arc — but the chord's own straight-line length never changes between them. Only the curved distance around the circle differs, depending on which way you travel.
Chord vs Diameter
A diameter is simply the special case of a chord that passes through the centre — it's the longest possible chord in any circle, always equal to 2r. Every other chord is strictly shorter than the diameter, which is exactly why this calculator rejects any chord length greater than 2r as geometrically impossible.
Chord vs Arc
These measure fundamentally different things. The chord is the straight-line distance between two points; the arc is the curved distance between the same two points, following the circle's edge. Since a straight line is always the shortest path between two points, the chord is always shorter than either arc it's associated with — the two only become nearly equal as the central angle shrinks toward zero.
Sagitta Explained
The sagitta (Latin for "arrow") is the height of the bulge between a chord and its arc — the perpendicular distance from the midpoint of the chord to the midpoint of the arc. It's a genuinely useful measurement in its own right: opticians and lens-makers use it to describe how curved a lens surface is, and engineers use it to describe how much a curved beam or arch rises above a straight baseline.
h = r − √(r² − (c/2)²) | c = 2√(2rh − h²)
Chord Formula
c = 2r × sin(θ/2)
The chord length is twice the radius times the sine of half the central angle. A circle with radius 10 and a 60° central angle has a chord of 2×10×sin(30°) = 2×10×0.5 = 10 units — in this particular case, the chord happens to equal the radius, which is a neat property of exactly 60°.
Distance from Centre
d = √(r² − (c/2)²)
Every chord sits at some perpendicular distance from the circle's centre — the shorter the chord, the further from the centre it typically sits; the longer the chord, the closer to the centre, until a diameter passes exactly through it (distance = 0). This distance, combined with the radius, is a reliable alternative way to specify a chord if you don't know the central angle directly.
Circular Segment
A circular segment is the region between a chord and its arc — imagine slicing a circle with a straight cut; the smaller piece (for a minor arc) is a minor segment, and the larger piece is a major segment. Segment area is genuinely different from sector area: a sector includes the two straight radii as well as the arc, while a segment only has the chord and the arc as its boundary.
Segment area = ½r²(θ − sinθ)
Worked Examples
Radius 10, angle 60°
Chord = 2×10×sin(30°) = 10 units; sagitta ≈ 1.340; distance from centre ≈ 8.660
Radius 10, angle 120°
Chord = 2×10×sin(60°) ≈ 17.321 units; sagitta = 5; distance from centre = 5
Radius 10, chord 10 (solve angle)
θ = 2×arcsin(10/20) = 60° — the minor angle; the reflex angle is 300°
Radius 10, distance from centre 8 (solve chord)
Chord = 2√(100−64) = 2×6 = 12 units
Chord 16, sagitta 4 (solve radius)
r = 16²/(8×4) + 4/2 = 8 + 2 = 10 units
Diameter example — radius 7, angle 90°
Chord = 2×7×sin(45°) ≈ 9.899 units — notably shorter than the diameter (14 units), confirming a 90° chord is not the diameter
Impossible-input example
Radius 10, chord entered as 25: a chord 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 | Arch and truss chord lengths, curved beam design, bridge span geometry |
| Mechanical engineering | Gear tooth chord measurements, pulley belt contact geometry |
| Optics | Lens curvature described by sagitta, mirror surface specifications |
| Manufacturing | Circular segment cutting, sheet-metal curved panel dimensions |
| Civil engineering | Culvert and pipe segment sizing, curved retaining wall panels |
Surveying Applications
| Use | How chord length helps |
|---|---|
| Road curve staking | Chords are used to lay out circular road curves in the field, since a straight tape measure can only measure chord distances directly, not the curve itself |
| Boundary surveys | Curved property boundaries are recorded as a chord bearing and distance, plus a radius and arc length |
| Tunnel and pipeline alignment | Chord offsets from a theoretical centreline confirm actual curved alignment matches the design |
| Large-radius curves | For gentle curves (a large radius, small angle), the chord and arc length are nearly identical — useful for quick field estimates |
Glossary
- Chord
- A straight line segment connecting two points on a circle's circumference.
- Diameter
- The longest possible chord — one that passes through the centre, equal to 2r.
- Sagitta
- The perpendicular height between the midpoint of a chord and the midpoint of its arc; also called segment height.
- Distance from centre
- The perpendicular distance from the circle's centre to the chord line.
- Central angle
- The angle at the centre between the two radii drawn to the chord's endpoints.
- Minor arc
- The shorter of the two arcs between the chord's endpoints (central angle strictly less than 180°).
- Major arc
- The longer of the two arcs between the same endpoints (central angle strictly greater than 180°).
- 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.
Common Mistakes
| Mistake | Fix |
|---|---|
| Confusing chord length with arc length | The chord is a straight line; the arc follows the curve — the arc is always the longer of the two |
| Assuming a chord is the same as the diameter | Only the chord passing exactly through the centre is a diameter; every other chord is shorter |
| Mixing degrees and radians | Convert consistently: degrees × π/180 = radians, radians × 180/π = degrees |
| Forgetting the chord + radius mode gives the minor angle only | The major (reflex) angle is always 360° − θ, and it shares the exact same chord |
| Confusing sagitta with distance from centre | They're related but different: sagitta = radius − distance from centre, not the same value |
Formula Cheat Sheet
Quick Reference
Chord:c = 2r×sin(θ/2) | Radius (from chord+angle): r = c/(2×sin(θ/2))Angle (from chord+radius):
θ = 2×arcsin(c/2r) | Distance from centre: d = √(r²−(c/2)²)Sagitta:
h = r−d | Chord from sagitta: c = 2√(2rh−h²) | Radius from chord+sagitta: r = c²/(8h)+h/2Minor arc:
L = rθ | Sector area: A = ½r²θ | Segment area: A = ½r²(θ−sinθ)
Practice Questions
Beginner (with answers)
- Find the chord length for radius 8 and central angle 90°.
- Find the chord length for radius 6 and central angle 60°.
- A circle has radius 5. What is the longest possible chord?
- Find the sagitta for radius 10 and central angle 90°.
- Is a chord ever longer than the diameter?
Show answers
1) 2×8×sin(45°)≈11.314 2) 2×6×sin(30°)=6 3) The diameter, 10 units 4) 10−10×cos(45°)≈2.929 5) No — the diameter is always the longest possible chord
Advanced (with answers)
- A chord of 14 has radius 10. Find the central angle in degrees.
- A chord of 12 and central angle 70° — find the radius.
- Find the distance from the centre for radius 13, chord 24.
- A chord is 18 with sagitta 3. Find the radius.
- Find the segment area for radius 10, central angle 90°.
Show answers
1) θ=2×arcsin(0.7)≈88.85° 2) r=12/(2×sin35°)≈10.46 3) d=√(169−144)=5 4) r=18²/(8×3)+3/2=13.5+1.5=15 5) A=½×100×(π/2−1)≈28.54
🔑 Key Takeaways
- A chord is the straight-line distance between two points on a circle: c = 2r×sin(θ/2).
- The diameter is the longest possible chord — every other chord is strictly shorter.
- The same chord defines two arcs (minor and major) but only one chord length — never both.
- Sagitta and distance from centre are two closely related, genuinely useful alternative measurements.
- This calculator offers eight starting combinations because chord problems arise from very different known values in practice — angle, sagitta, or a direct centre-distance offset.
Frequently Asked Questions
A straight line segment connecting two points on a circle's circumference. Unlike a radius, it doesn't need to pass through the centre.
c = 2r×sin(θ/2), using the radius and the central angle between the chord's two endpoints. A circle with radius 10 and a 60° angle has a chord of exactly 10 units.
c = 2r×sin(θ/2), where r is the radius and θ is the central angle in radians (or converted from degrees first).
If you know the central angle: r = c/(2×sin(θ/2)). If you know the sagitta instead: r = c²/(8h) + h/2.
θ = 2×arcsin(c/2r), using the chord length and radius. This always gives the minor angle; the major (reflex) angle is 360° minus that result.
A diameter is the special case of a chord that passes exactly through the centre — the longest possible chord in any circle, always equal to 2r. Every other chord is shorter.
No — the diameter is always the longest possible chord in a circle. Any chord length greater than 2r is geometrically impossible, and this calculator rejects it.
The chord is the straight-line distance between two points on a circle; the arc is the curved distance between the same two points, following the circle's edge. The chord is always shorter than the arc.
Two — a minor arc and a major arc, on either side of the chord. The chord's own length stays exactly the same regardless of which arc you're considering; only the two arc lengths differ.
The perpendicular height from the midpoint of the chord to the midpoint of its arc — literally "arrow" in Latin, describing how far the arc bulges away from the straight chord.
h = r − √(r² − (c/2)²), using the radius and chord. Equivalently, h = r×(1 − cos(θ/2)) if you know the central angle instead.
c = 2√(2rh − h²), using the radius and sagitta together.
r = c²/(8h) + h/2 — a direct rearrangement that needs no trigonometry, which is why sagitta measurements are popular in optics and manufacturing.
The perpendicular distance from the circle's centre to the chord line: d = √(r² − (c/2)²). It's zero for a diameter and increases as the chord gets shorter.
They add up to the radius: h + d = r. Knowing either one, plus the radius, gives you the other directly.
The region between a chord and its arc, not including the two radii. It's different from a sector, which does include the two straight radii as part of its boundary.
A = ½r²(θ − sinθ), using the central angle in radians. This equals the sector area minus the triangle formed by the two radii and the chord.
A sector is bounded by two radii and an arc — like a pizza slice. A segment is bounded only by a chord and an arc, with no radii — like the piece cut off when you slice straight across a circle.
The minor arc is the shorter path between two points on a circle (central angle under 180°); the major arc is the longer path (central angle over 180°). Both share the exact same chord.
Because the same chord always corresponds to two different arcs with two different lengths and areas — showing only one would silently pick a side of a genuinely two-way answer.
Yes — this is the most direct mode: c = 2r×sin(θ/2), giving an exact answer with no ambiguity.
Yes — this calculator's Diameter + Angle mode simply halves the diameter to get the radius first, then applies the usual chord formula.
Yes — c = 2√(r² − d²), using the radius and the perpendicular distance from the centre to the chord.
Any linear unit for the radius, chord, sagitta and distance from centre (mm, cm, m, ft); sector and segment area use the squared version of the same unit; the central angle is in degrees or radians with no length unit.
Yes — choose any of the eight modes above, enter the known values, and the calculator solves everything else instantly with the working shown.
All eight modes on this calculator are solved with exact closed-form formulas, accurate to floating-point precision — no approximation is needed for any chord calculation.
Yes — enter the known values for whichever mode matches, and the calculator shows the formula and full working, useful for checking homework as well as producing an answer.
Confusing chord length with arc length; assuming any chord is the diameter; mixing degrees and radians; and confusing sagitta with distance from centre, which are related but not the same value.
Arch and truss design, curved beam geometry, and bridge span calculations all rely on chord length alongside radius and sagitta to define the curve precisely.
Lens and mirror curvature is commonly specified using the sagitta over a known chord (aperture) diameter — a direct, trigonometry-free way to describe how curved an optical surface is.
Road curves are staked out in the field using chord lengths, since a tape measure can only measure straight-line distances directly — the curve itself is confirmed indirectly through a series of chord offsets.
Circular segment cutting and curved sheet-metal panels are dimensioned using chord and sagitta, since these translate directly into straightforward material layout measurements.
The calculator uses the required values for the selected mode to solve the chord, and cross-checks any extra values you entered (minor arc length or sector area) against the result, flagging a warning if they don't match.
Yes — use Copy Results to copy everything to the clipboard, Copy Formula for just the formulas used, Export CSV for a spreadsheet-ready file, Download SVG Diagram for a vector image, or Print / Save as PDF for a printable worksheet.
A chord 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 chord problems arise from very different starting values in practice — sometimes an angle, sometimes a sagitta measurement, sometimes a direct centre-distance offset.
The calculator rejects it — a sagitta can never exceed the diameter, since the maximum possible bulge from a chord to its arc is bounded by the full width of the circle.
The calculator rejects it — the perpendicular distance from the centre to any chord can never exceed the radius itself, since the chord has to actually touch the circle.
No — a chord needs a positive central angle greater than 0° and no more than 360°; this calculator validates that range and explains why if it's violated.
There's no fixed minimum size — any positive radius and any central angle between 0° and 360° describe a valid chord, approaching zero length as the angle approaches zero.
No — chord length scales directly with radius for the same angle (c = 2r×sin(θ/2)), so a 60° chord in a radius-20 circle is exactly twice as long as a 60° chord in a radius-10 circle.
The point exactly halfway along the chord — it always lies on the line from the circle's centre through the sagitta, perpendicular to the chord itself.
Yes — in a given circle, equal chords are always equidistant from the centre, and conversely, chords equidistant from the centre are always equal in length. This is a direct consequence of the same formula applying to both.
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.
- All eight solve modes have exact closed-form solutions — no numerical approximation is used anywhere on this page.
- The Chord + Radius mode always returns the minor central angle; the major (reflex) angle and its arc are shown separately in the comparison panel.
About This Calculator
Printable Formula Sheet
A one-page reference with every chord formula on this page.
Chord Length Formula Sheet
MegaCalcOnline.com · Chord, sagitta, distance and segment formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Chord length | c = 2r×sin(θ/2) |
| Radius (from chord + angle) | r = c/(2×sin(θ/2)) |
| Central angle (from chord + radius) | θ = 2×arcsin(c/2r) |
| Distance from centre | d = √(r²−(c/2)²) |
| Sagitta | h = r−d = r(1−cos(θ/2)) |
| Chord from sagitta | c = 2√(2rh−h²) |
| Radius from chord + sagitta | r = c²/(8h) + h/2 |
| Minor arc length | L = rθ |
| Sector area | A = ½r²θ |
| Segment area | A = ½r²(θ−sinθ) |
References
- Khan Academy — Chords, arcs and circle geometry
- OpenStax Mathematics — circular measurement
- Wolfram MathWorld — Chord
- NIST — SI units for length and area
- Maths Is Fun — Circle Sector and Segment
Every formula and worked example on this page was independently verified — see “About This Calculator” above for the full review notes.