Unit Circle Calculator — Trig Values, Coordinates & Quadrants
Enter any angle in degrees or radians to find sin, cos, tan, csc, sec, cot, (x, y) coordinates, quadrant, and reference angle on the unit circle. Common angles table included.
Chord Length Calculator
Calculate chord length, arc length, sagitta, segment area, and sector area from radius and central angle, with circle diagram, reference table, and visual comparison bars.
Chord Length⧉
10.000000
c = 2r·sin(θ/2) — straight-line distance between chord endpoints
Arc Length⧉
10.471976
s = r·θ — length of the curved arc subtended by the angle
Sagitta (Height)⧉
1.339746
h = r(1 − cos(θ/2)) — max distance from chord to arc
Apothem⧉
8.660254
d = r − h — distance from center to chord midpoint
Segment Area⧉
9.058607
A_seg = r²(θ − sin θ)/2 — area between chord and arc
Sector Area⧉
52.359878
A_sec = r²θ/2 — area of the pie-slice sector
Triangle Area⧉
43.301270
Area of triangle from center to chord endpoints
Central Angle (°)⧉
60.000000°
The angle at the center subtended by the chord
Circle Diagram
Chord Properties by Common Angle (r = 10.00)
| Angle | Chord | Arc | Sagitta | Segment Area |
|---|---|---|---|---|
| 30° | 5.1764 | 5.2360 | 0.3407 | 1.1799 |
| 45° | 7.6537 | 7.8540 | 0.7612 | 3.9146 |
| 60° | 10.0000 | 10.4720 | 1.3397 | 9.0586 |
| 90° | 14.1421 | 15.7080 | 2.9289 | 28.5398 |
| 120° | 17.3205 | 20.9440 | 5.0000 | 61.4185 |
| 135° | 18.4776 | 23.5619 | 6.1732 | 82.4544 |
| 150° | 19.3185 | 26.1799 | 7.4118 | 105.8997 |
| 180° | 20.0000 | 31.4159 | 10.0000 | 157.0796 |
Chord Length as % of Diameter
30°
25.9%
45°
38.3%
60°
50.0%
90°
70.7%
120°
86.6%
135°
92.4%
150°
96.6%
180°
100.0%
Planning notes, formulas, and examples
About the Chord Length Calculator
A chord is a straight line segment whose endpoints both lie on a circle. The chord length formula — c = 2r sin(θ/2) — connects the radius r and the central angle θ to the straight-line distance between the two points where the chord meets the circle. It is one of the most frequently used relationships in circle geometry, appearing in fields from civil engineering (road curves, tunnel cross-sections) to optics (lens apertures) and music theory (string vibration).
Beyond the chord itself, this calculator computes every related quantity: the arc length s = rθ (the curved distance along the circle between the endpoints), the sagitta h = r(1 − cos θ/2) (the maximum perpendicular distance from the chord to the arc — also called the "versed sine" or arc height), the apothem (distance from the center to the chord midpoint), the segment area (the region between the chord and the arc), the sector area (the pie-slice region), and the triangle area formed by the two radii and the chord.
Three input modes cover the most common practical scenarios. You can enter the radius and the central angle directly, or supply the radius and the known chord length to back-solve for the angle, or provide the radius and the sagitta. Preset buttons load six popular configurations, and a reference table lists all these measurements for eight standard angles (30° through 180°) at your entered radius. The circle diagram visually labels the chord, sagitta, sector, and radii so you can relate the numbers back to the geometry.
When This Page Helps
Chord and segment calculations require trigonometric identities involving half-angles and inverse trig functions — easy to mix up, especially when switching between degrees and radians. This calculator handles all three input scenarios (radius+angle, radius+chord, radius+sagitta), computes eight related quantities at once, and draws an SVG diagram labelling every measurement. Civil engineers sizing road curves, surveyors measuring arc segments, and students checking geometry homework all get reliable answers without manual trig.
How to Use the Inputs
- Select the input mode: Radius + Angle, Radius + Chord Length, or Radius + Sagitta.
- Choose degrees or radians for the angle unit.
- Enter the radius of the circle.
- Enter the second input (angle, chord length, or sagitta) depending on your mode.
- Read all eight output values — chord, arc, sagitta, apothem, areas, and angle.
- View the SVG circle diagram showing the chord, sector, and sagitta.
- Consult the reference table for the same radius at eight standard angles.
- Use the bar chart to visualize how chord length varies as a fraction of the diameter.
Formula used
Chord length: c = 2r·sin(θ/2)
Arc length: s = r·θ
Sagitta: h = r·(1 − cos(θ/2))
Apothem: d = r − h = r·cos(θ/2)
Segment area: A_seg = r²·(θ − sin θ) / 2
Sector area: A_sec = r²·θ / 2Example Calculation
Result: Chord = 10, Arc ≈ 10.472, Sagitta ≈ 1.340
c = 2·10·sin(30°) = 20·0.5 = 10. Arc = 10·(π/3) ≈ 10.472. Sagitta = 10·(1 − cos 30°) ≈ 10·0.134 = 1.340.
Tips & Best Practices
- A chord spanning exactly 60° of a circle equals the radius — a useful quick-check.
- The longest possible chord is the diameter (central angle = 180°).
- Sagitta is useful in optics for calculating lens curvature from chord and arc measurements.
- Double-check that your calculator is in the correct angle mode (degrees vs. radians).
- For small angles, chord ≈ arc length — the difference becomes significant only for large angles.
Deriving the Chord Length Formula
Draw two radii to the endpoints of the chord, creating an isosceles triangle with two sides equal to r and the included angle equal to the central angle θ. Dropping a perpendicular from the centre to the chord bisects both the chord and the angle. In the resulting right triangle, sin(θ/2) = (c/2) / r, so c = 2r sin(θ/2). This elegant result connects a straight-line measurement to a circular angle and is the starting point for every other formula on this page — arc length, sagitta, segment area, and sector area are all derived from r and θ.
Engineering and Surveying Applications
Chord geometry is everywhere in infrastructure design. Road curves are specified by radius and deflection angle; surveyors stake out the curve by calculating chord distances between successive points. Railway engineers use chord lengths to set super-elevation on bends. In tunnel construction, a circular cross-section is cut by measuring chords at key heights (essentially the sagitta in reverse). Optical engineers use the sagitta formula to measure lens curvature from the chord across the aperture. Even in everyday life, bending a strip of metal into an arc requires knowing the chord-to-arc ratio to predict spring-back.
Segment and Sector Area: Knowing the Difference
The sector area (r²θ/2) is the "pie-slice" between two radii and the arc — it includes the triangular area from the centre. The segment area is the region between the chord and the arc, computed as sector minus triangle: A_seg = r²(θ − sin θ)/2. This distinction matters in fluid mechanics (cross-sectional area of liquid in a partially filled pipe), architecture (stained-glass window panels), and volume calculations for horizontal cylindrical tanks. The calculator provides both areas so you can pick the one relevant to your problem without extra arithmetic.
Sources & Methodology
Last updated:
Frequently Asked Questions
A chord is a straight line segment connecting two points on the circumference of a circle. The diameter is the longest chord.
The sagitta (also called "arc height" or "versed sine") is the maximum perpendicular distance from the midpoint of the chord to the arc.
The sector area is the pie-slice between two radii and the arc. The segment area is the region between the chord and the arc (sector minus the triangle).
Yes — use the "Radius + Chord Length" mode. The calculator back-solves θ = 2·arcsin(c / 2r).
The chord formula still works for reflex angles (180° < θ < 360°); the chord connects the same two points but the arc goes the long way around.
Yes — the straight line (chord) is always shorter than or equal to the curved path (arc). They are equal only when the angle is 0.
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