Quick answer
A chord subtending a central angle θ in a circle of radius r has length 2r sin(θ ÷ 2). With r = 5 and θ = 60°, the chord is 5.
Enter the radius and central angle. The calculator returns the chord length.
Related tools: Arc Length Calculator: Length of a circular arc. Diagonal Length Calculator: Diagonal of a rectangle or the space diagonal of a box. Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides.
How to use the Chord Length Calculator
- Enter radius (above 0).
- Enter central angle (°) (between 0 and 180).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Radius | A number | > 0 |
| Central angle | A number, in ° | > 0, ≤ 180 |
Formula and method
Formula
chord = 2r·sin(θ ÷ 2)
IEEE-754 double precision; shown to at most 6 decimal places.
Isosceles-triangle geometry; angle limited to 180° (a diameter).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Radius = 5; Central angle = 60.
| Result | Value |
|---|---|
| Chord length | 5 |
Example 2
Inputs: Radius = 5; Central angle = 180.
| Result | Value |
|---|---|
| Chord length | 10 |
Example 3
Inputs: Radius = 10; Central angle = 90.
| Result | Value |
|---|---|
| Chord length | 14.142136 |
The first line is the headline result (chord length). IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
A 60° chord forms an equilateral triangle with two radii, so the chord equals the radius. 2 × 5 × sin 30° = 5. At 180° the chord is the diameter, 10.
Related ideas
The sagitta (height of the arc) is r(1 − cos(θ/2)). For a chord c and radius r the angle is 2 arcsin(c ÷ 2r). The angle must be between 0° and 180° for the shorter arc.
How the result changes with radius
Holding every other input at the first worked example, this table changes only radius. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Radius | Chord length |
|---|---|
| 2.5 | 2.5 |
| 5 | 5 |
| 10 | 10 |
| 20 | 20 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Central angle: entering "200" shows "Central angle must be at most 180." beside that field and no result.
Choosing between the Chord Length Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Arc Length Calculator](/math/arc-length-calculator/) | radius and central angle | Arc length |
| [Diagonal Length Calculator](/math/diagonal-length-calculator/) | length and width | Diagonal |
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [Circle Calculator](/math/circle-calculator/) | i know the and value | Area |
| [Circumference Calculator](/math/circumference/) | i know the and value | Circumference |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Angle 180° gives the diameter
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for central angle to confirm this.
Frequently asked questions
What is a chord?
A line segment joining two points on a circle.
What's the longest chord?
The diameter, which is the chord through the centre (the 180° case, length 2r).
How do I find the angle from a chord?
Use θ = 2 arcsin(c ÷ 2r); a chord of 5 in a circle of radius 5 gives 60°.
What does the Chord Length Calculator calculate?
Length of a chord from the radius and central angle. It returns Chord length.
What formula does the Chord Length Calculator use?
chord = 2r·sin(θ ÷ 2). Isosceles-triangle geometry; angle limited to 180° (a diameter).
What do I need to enter in the chord Length Calculator?
You enter Radius and Central angle (°).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Angle 180° gives the diameter.
What is an example of a chord Length result?
With radius 5 and central angle 60, the chord length is 5.
Related calculators
- Arc Length Calculator — Length of a circular arc.
- Diagonal Length Calculator — Diagonal of a rectangle or the space diagonal of a box.
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
- Circumference Calculator — Circumference of a circle.
- Area of a Circle Calculator — Area of a circle from its radius, diameter or circumference.
- Hexagon Calculator — Area, perimeter and angles of a regular 6-sided polygon.
- Pentagon Calculator — Area, perimeter and angles of a regular 5-sided polygon.
- Octagon Calculator — Area, perimeter and angles of a regular 8-sided polygon.
Sources
- Standard Euclidean geometry formulas (e.g. NIST Digital Library of Mathematical Functions §4, CRC Standard Mathematical Tables)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.