Chord Length Calculator: How to Calculate the Chord of a Circle
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Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
A chord length calculator solves a problem that shows up constantly in construction, engineering, and design: you know a circle's radius and one other measurement, and you need the straight-line distance between two points on the curve. Say you're building a curved retaining wall with a 15-meter radius and the client wants a 4-meter opening. Can that chord fit a 90° arc? Or suppose you're cutting a pipe along a curve and need the flat distance across the cut. The formula is short — 2r sin(θ/2) — but the real challenge is picking the right inputs and avoiding unit errors that silently ruin the result.
The Geometry That Makes Chord Formulas Work
Every chord in a circle creates an isosceles triangle when you draw radii to each endpoint. The two equal sides are the radii (length r), and the base is the chord itself. Drop a perpendicular from the center to the chord and you split everything into two right triangles. That perpendicular bisects both the chord and the central angle — and that's where the formula comes from.
In each right triangle, the hypotenuse is r, the angle at the center is θ/2, and the opposite side (half the chord) is r × sin(θ/2). Double it and you get:
Chord = 2r × sin(θ/2)
where r = radius, θ = central angle in radians
This single identity is the backbone. Every other chord formula — from sagitta, from perpendicular distance, from arc length — is just an algebraic rearrangement of the same triangle.
Four Ways This Chord Length Calculator Finds Your Answer
You rarely get to choose your measurements. A surveyor might have the arc height but not the angle. An architect knows the perpendicular clearance but not the sagitta. Here are the four routes, each starting from what you actually have.
| You Have | Formula | When to Use |
|---|---|---|
| r + θ (central angle) | 2r sin(θ/2) | CAD drawings, protractor readings, math problems |
| r + h (sagitta) | 2√(h(2r − h)) | Arch construction, dome measurements, road crowns |
| r + d (distance to center) | 2√(r² − d²) | Pipe offsets, tunnel clearances, nearest-point calculations |
| r + s (arc length) | θ = s/r, then 2r sin(θ/2) | Cable routing, belt drives, GPS great-circle shortcuts |
Worked Example: Finding the Span of a Stone Arch
A mason is building a semicircular stone arch with a 3-meter radius. The arch subtends a 120° central angle (not a full semicircle — the walls take the rest). What is the straight-line span across the opening?
Given: r = 3 m, θ = 120°
Step 1: Convert angle → 120° × π/180 = 2.0944 rad
Step 2: Half-angle → 2.0944/2 = 1.0472 rad
Step 3: sin(1.0472) = 0.86603
Step 4: Chord = 2 × 3 × 0.86603 = 5.196 m
The opening is about 5.2 meters wide — roughly the width of a single-lane road.
Notice that 120° gives a chord equal to r√3, one of those clean results that shows up in hexagonal geometry. If you need to double-check your answers, our chord calculator page covers the same formulas from a different angle with additional reference tables.
Worked Example: Road Crown from Sagitta
A civil engineer measures that a two-lane road has a crown (sagitta) of 0.15 m across its 7-meter width. She needs the radius of the road's cross-sectional curve. But first, let's verify: given r and sagitta, what chord do we get? Suppose after surveying she finds r = 41.04 m and h = 0.15 m.
Given: r = 41.04 m, h = 0.15 m
Formula: Chord = 2√(h(2r − h))
Step 1: 2r − h = 2(41.04) − 0.15 = 81.93
Step 2: h × 81.93 = 0.15 × 81.93 = 12.2895
Step 3: √12.2895 = 3.5056
Step 4: Chord = 2 × 3.5056 = 7.011 m
That matches the 7-meter road width — the sagitta formula checks out.
Road crowns are a real application of the sagitta-chord relationship. Civil engineers also use it for railway superelevation and drainage pipe curvature. The perpendicular distance variant (2√(r² − d²)) is essentially the same idea rotated: the distance from center replaces the sagitta.
Why the Arc Is Always Longer Than the Chord
One of the most useful sanity checks: the chord is always shorter than the arc connecting the same two points (unless both are zero). At small angles the difference is negligible — a 10° arc on a circle of radius 100 gives an arc of 17.453 and a chord of 17.431, a gap of barely 0.1%. But at 180°, the chord is the diameter (2r) while the arc is πr, making the arc about 57% longer.
This ratio matters in arc length calculations for navigation. GPS devices calculate great-circle arcs, but a pilot's straight-line distance is the chord. For a 1,000 km great-circle route at Earth's surface, the chord through the Earth is shorter — and the discrepancy grows with distance.
Three Mistakes That Silently Break Your Results
1. Degrees vs. radians mix-up
Plugging 60 (degrees) into 2r sin(θ/2) without converting gives 2r sin(30) = 2r × (−0.988) — a negative number for a length. The formula expects radians. 60° is π/3 ≈ 1.0472 radians. Always check your units before computing.
2. Sagitta larger than the diameter
The sagitta formula 2√(h(2r − h)) produces imaginary numbers when h > 2r. Physically this means the "arc height" exceeds the full diameter, which is impossible. If your sagitta is larger than the diameter, you've measured the wrong distance.
3. Using the inscribed angle instead of the central angle
The inscribed angle (from a point on the circle) is exactly half the central angle (from the center). If you accidentally use an inscribed angle of 60° as if it were the central angle, your chord will be too short by a factor of sin(30°)/sin(60°) ≈ 0.577. When a problem says "angle subtended by the chord," ask: from the center or from the circumference?
Chord Length Reference for a Unit Circle (r = 1)
This table gives exact and decimal chord lengths for a unit circle. To find the chord for any radius, just multiply by your radius. For instance, on a circle with r = 25, the chord at 90° is 25 × 1.4142 = 35.355.
| θ | Exact Chord | Decimal | Chord/Arc |
|---|---|---|---|
| 30° | 2 sin 15° | 0.5176 | 0.9886 |
| 60° | 1 | 1.0000 | 0.9549 |
| 90° | √2 | 1.4142 | 0.9003 |
| 120° | √3 | 1.7321 | 0.8270 |
| 150° | 2 cos 15° | 1.9319 | 0.7380 |
| 180° | 2 (diameter) | 2.0000 | 0.6366 |
The Chord/Arc column shows how the straight-line shortcut compares to following the curve. At 60° the chord is about 95.5% of the arc; at 180° it drops to 63.7% — the well-known 2/π ratio. This ratio appears in physics when comparing a pendulum's arc displacement to its horizontal displacement.
Beyond Flat Circles: Chords on Spheres and Cylinders
The chord formula extends directly to spherical geometry. On Earth (radius ≈ 6,371 km), the chord between two cities follows the same 2R sin(θ/2) pattern — just using the great-circle central angle. The chord between New York and London (great-circle angle ≈ 51.4°) works out to about 5,564 km, while the surface arc distance is 5,570 km. For most practical purposes the difference is under 1%.
In machining, chords on cylinders determine how many flat facets you need to approximate a curve within a given tolerance. If a CNC machine cuts flat segments along a 100 mm radius at 5° increments, each chord is 2 × 100 × sin(2.5°) = 8.716 mm, and the maximum deviation (sagitta) from the true curve is just 0.095 mm. You can explore more circle properties and how they relate to sector area calculations on their dedicated pages.
External reference: For the mathematical proofs behind chord-angle relationships and their extension to inscribed angle theorems, see the Wikipedia article on chords.
