Library / Geometry, coordinates, surveying and trigonometry / GEO-06
Geometry, coordinates, surveying and trigonometry
Chord length from the central angle
Length of a chord from the radius and the central angle it subtends.
- Ref
- GEO-06
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- c [cm]
A chord joins two points on a circle. Drawing the radii to its ends gives an isosceles triangle with apex angle θ. Splitting it down the middle gives two right triangles with hypotenuse r and angle θ/2, in which half the chord is r sin(θ/2). So c = 2r sin(θ/2). At θ = π the chord is a diameter.
Length of a chord from the radius and the central angle it subtends. Bisecting the chord from the centre gives a right triangle with hypotenuse r.
Calculator
- Distance from the centre to the chord, r cos(θ/2)4.33 cm
- Length of the minor arc, rθ5.236 cm
- Central angle in degrees60 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 60° of arc on a 5 cm radius the chord is 5 cm, against an arc of 5.236 cm.
- For a 60° angle (1.0472 rad) the chord equals the radius, because the triangle is equilateral.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Radius | 1 to 30 cm | 1 | 30 | 29 | +10% | +10 % |
| θ Central angle (the smaller, 0 to π) | 0.05 to 3.142 rad | 0.25 | 10 | 9.75 | +10% | +8.93 % |
Sweep of Central angle (the smaller, 0 to π)
| Central angle (the smaller, 0 to π) (rad) | Chord length (cm) |
|---|---|
| 0.05 | 0.25 |
| 0.4364 | 2.165 |
| 0.8229 | 3.999 |
| 1.209 | 5.685 |
| 1.596 | 7.159 |
| 1.982 | 8.366 |
| 2.369 | 9.263 |
| 2.755 | 9.814 |
| 3.142 | 10 |
Sweep one variable
Variables and how each one contributes
Chord length
What the equation gives you. Everything else feeds it.
Radius
The radius of the circle, in cm. A larger radius gives a longer chord for the same angle, in direct proportion. Raising it by 10 % (to 5.5) moves the result by +10 %, from 5 to 5.5 cm.
Central angle (the smaller, 0 to π)
The angle at the centre between the two radii, in radians, from 0 to π. A wider angle gives a longer chord, up to the diameter at π. Raising it by 0.05 (to 1.097) moves the result by +4.3 %, from 5 to 5.215 cm.
Worked example with the values above
- Half the angle: θ/2 = 1.047 / 2 = 0.5236 rad.
- Sine: sin(0.5236) = 0.5.
- c = 2 × 5 × 0.5 = 5 cm.
Assumptions and limits
- A true circle, with angles in radians where the formula uses them.
- Lengths are measured from the same centre, in one consistent unit.
Common mistakes
- Entering a central angle in degrees where radians are needed.
- Using the full chord in place of the half chord when working with the right triangle.
- Using the major arc when the minor arc is meant.
Statistics and uncertainty
Circle results depend on the radius squared or on trigonometric functions of the angle, so errors in the radius are doubled in areas. The segment and rise formulas can lose digits for very small angles or very shallow arcs.
Where it is used
- Cutting, rolling and setting out curved plates, pipes and arches.
- Partly filled pipes and tanks, washers and flanges, and tangent lines past a circular object.
Maths you need
- Radians: angle measured as arc length over radius
- Right triangle: Pythagoras and sine over hypotenuse
- Circle area: π times radius squared