Chord Length: From the Formula to Cutting a Real Curve

What a chord actually is

A chord is any straight line whose two ends both sit on a circle. That is the whole definition — no need for it to pass through the centre, no need for it to be special in any way. The two longest chords you can draw are the diameters, each running clean through the middle; the shortest are the tiny near-zero ones connecting two points a hair apart. Everything in between is still a chord.

The reason chords get their own formula is that a circle is so regular. Fix the radius and the chord’s length depends on exactly one thing: how far apart its two endpoints are around the rim, measured as the angle they make back at the centre. That angle is the central angle, written θ, and it is the single input that turns a radius into a chord. The chord length calculator takes those two numbers — radius and central angle — and returns the chord instantly, along with the arc, the sagitta and the distance from the centre. This guide is about where that formula comes from, how to use it when the angle isn’t the thing you actually know, and the places chords quietly turn up in real work.

The chord length formula, and why it looks like that

The headline formula is short:

c = 2 · r · sin(θ/2)

where r is the radius and θ is the central angle. The θ/2 and the factor of 2 out front are the parts worth understanding, because once you see why they are there the formula stops being something to memorise.

Draw the two radii from the centre out to the chord’s endpoints. Together with the chord they enclose an isosceles triangle — two equal sides of length r, with the angle θ between them. Now drop a line from the centre straight down to the midpoint of the chord. Because the triangle is isosceles, that line splits it perfectly in half: it bisects the angle θ into two halves of θ/2, and it bisects the chord into two halves. What you are left with is a right triangle whose hypotenuse is r and whose angle at the centre is θ/2. In that triangle the side opposite the angle is half the chord, so half the chord is r·sin(θ/2). Double it and you have the whole thing: c = 2·r·sin(θ/2).

The same right triangle hands you two more measurements for free. The side adjacent to the θ/2 angle is the perpendicular distance from the centre down to the chord, d = r·cos(θ/2) — sometimes called the apothem. And the gap between that distance and the full radius is the sagitta, h = r − r·cos(θ/2), the height of the arc bulging above the chord. The arc itself, the curved piece the chord cuts off, has length s = r·θ once θ is in radians. All four numbers come out of one split triangle, which is why the calculator can report them from a single pair of inputs.

Worked example: a quarter-turn chord

Take a circle of radius 5 and a chord that subtends a central angle of 90° — a right angle at the centre. Half the angle is 45°, so:

c = 2 · 5 · sin(45°) = 10 · 0.707107 = 7.071068

That is 5√2, in the same length unit as the radius. It is worth checking against the other route. The perpendicular distance from the centre to the chord is d = 5·cos(45°) = 3.535534, and the chord can equally be written c = 2·√(r² − d²) = 2·√(25 − 12.5) = 2·√12.5 = 7.071068 — the two formulas agree exactly, as they must. The arc that this chord cuts off is s = 5·(π/2) = 7.853982, a touch longer than the chord because it takes the curved path, and the sagitta is h = 5·(1 − cos 45°) = 1.464466.

Two landmark angles are worth carrying in your head as sanity checks. A 60° central angle gives c = 2·r·sin(30°) = r — the chord equals the radius exactly, because the two radii and the chord form an equilateral triangle. And a 180° angle gives c = 2·r·sin(90°) = 2r, the full diameter, which is as long as a chord can ever get. If you feed either of those into the chord length calculator and get anything else, the input is wrong.

Finding a chord when you don’t know the angle

Here is the catch that sends most people looking for a calculator in the first place: in real problems you almost never know the central angle. You know a width and a height, or a radius and a depth, and the angle is the hidden variable. Happily, the chord doesn’t care how you arrived at it.

From the radius and the distance to the centre

If you know the radius r and the perpendicular distance d from the centre of the circle to the chord, skip the angle entirely:

c = 2 · √(r² − d²)

This is just the Pythagorean theorem on that same split triangle — half-chord, distance and radius are the two legs and the hypotenuse of a right triangle, so (c/2)² + d² = r². A chord 3 units from the centre of a radius-5 circle is 2·√(25 − 9) = 2·4 = 8 units long.

From the radius and the sagitta

If instead you know the radius and the sagitta h — the height of the arc above the chord — then d = r − h, and the same formula becomes c = 2·√(r² − (r − h)²), which tidies up to:

c = 2 · √(2·r·h − h²)

A radius-5 arc with a sagitta of 1.464466 gives 2·√(2·5·1.4645 − 1.4645²) = 2·√(14.6447 − 2.1447) = 2·√12.5 = 7.071068 — back to our worked example, arrived at without ever mentioning 90°.

Going the other way: radius from a chord and a height

This is the version tradespeople reach for most. You can measure the width of an arch and the height of its curve, but the radius and the angle are both invisible. Rearranging the sagitta relationship gives the radius directly from the two things you can measure:

r = (c²/4 + h²) / (2·h)

Measure a curved window opening at 80 cm wide (c = 80) with a rise of 20 cm at the middle (h = 20), and the radius of that curve is (1600 + 400) / 40 = 2000 / 40 = 50 cm. This chord-and-sagitta pair is enough to reconstruct the entire circle the arc was cut from — a fact used constantly in joinery, sheet-metal work and stonemasonry, where the sagitta is the natural thing to measure with a straightedge and a ruler.

Where chord length actually shows up

Chords are one of those pieces of school geometry that turn out to be everywhere once you look.

Regular polygons inside circles

Every side of a regular polygon inscribed in a circle is a chord, and they all subtend the same central angle: 360° divided by the number of sides. A hexagon splits the circle into six 60° slices, so each side is a 60° chord — which, as we saw, equals the radius. That is exactly why a regular hexagon’s side length matches the circle it fits inside, and it generalises: a pentagon’s side is a 72° chord, an octagon’s a 45° chord. Drawing a polygon from a circle is nothing more than repeating one chord.

Arches, curves and fabrication

Anyone laying out a curve from flat stock — a bent handrail, an arched doorway, a curved worktop — works in chords and sagittas, because those are the measurements a tape and a square can reach. The width across is the chord, the rise at the middle is the sagitta, and the pair fixes the radius the curve has to be cut to. The sagitta also runs through optics, where it describes the depth of a spherical lens or mirror, and through surveying and road design, where chord-offset methods set out gentle horizontal curves.

Inscribed angles and the wider circle theorems

The central angle is only one angle that stands on a chord. From any point on the far arc, the same chord subtends an inscribed angle that is exactly half the central angle — the inscribed angle theorem. That relationship is the backbone of a whole family of circle results, and it is why a chord drawn as a diameter subtends a right angle everywhere on the circle (Thales’ theorem): the diameter is a 180° chord, so its inscribed angle is 90°.

Common mistakes

Halving the angle in the wrong place. The formula is 2·r·sin(θ/2), not 2·r·sin(θ). It is thehalf-angle that lives inside the sine, because the split triangle only contains θ/2. Plug the full angle into the sine and a 90° chord comes out as 2·5·sin(90°) = 10 — the diameter — instead of 7.07.

Feeding radians a degree, or vice versa. The chord formula needs sin(θ/2) evaluated correctly, and the arc formula s = r·θ needs θ in radians. Mixing the two is the classic slip: convert with θ_rad = θ_deg · π/180 before touching the arc. The calculator takes degrees and handles this conversion internally, which is precisely the trap it exists to remove.

Confusing the chord with the arc. The chord is the straight shortcut; the arc is the curved path between the same two points. The arc is always the longer of the two for any real chord, and the sagitta is the gap between them. Reaching for arc length when a problem wants the straight distance (or the reverse) is one of the most common errors on this topic.

Assuming a chord can beat the diameter. No chord is longer than 2r. If a calculation hands you a chord bigger than the diameter, the radius or the angle is wrong — the sine term maxes out at 1 when θ = 180°, and 2·r·1 = 2r is the ceiling.

When the chord is only the start

Chord length is rarely the final answer — it is usually a step towards laying out a shape, cutting a curve or unlocking the rest of a circle. Once you have the radius, the circle calculator takes you on to circumference and area, and if the circle is sitting in a coordinate plane the circle equation calculator writes down its equation. For the geometry that sits underneath all of this, the circle calculator guide and the Pythagorean theorem explainer cover the two tools every chord problem leans on. The definitions and the core derivation also live on the chord length calculator page itself. The treatment here follows MathWorld’s Chord entry and math.net’s sagitta page, both worth a bookmark for the formal statements.