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Chord Length Calculator

Calculate the length of a chord in a circle given the radius and the central angle subtended by the chord. Uses the formula c = 2r*sin(theta/2), essential for geometry, engineering, surveying, and arc-based constructions in architecture and design.

Reviewed by Chase FloiedUpdated

This free online chord length calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Radius of the circle

Central angle in degrees subtended by the chord

Results

Chord Length

10

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Chord Length Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Chord Length Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Chord Length Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Chord Length Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Chord Length Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the length of a chord in a circle given the radius and the central angle subtended by the chord. Uses the formula c = 2r*sin(theta/2), essential for geometry, engineering, surveying, and arc-based constructions in architecture and design. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Chord Length Calculator

The Chord Length Calculator computes the straight-line distance between two points on a circle using the radius and the central angle between them. The formula c = 2r*sin(theta/2) is derived from basic trigonometry of the isosceles triangle formed by two radii and the chord. Chord length calculations are fundamental in circle geometry and have practical applications in surveying (measuring distances along circular boundaries), civil engineering (designing curved roads and bridges), architecture (constructing arches and domes), and music theory (dividing a circular drum head). This calculator accepts the angle in degrees for convenience.

The Math Behind It

A chord is a line segment whose endpoints lie on a circle. The chord length formula c = 2r*sin(theta/2) comes from the isosceles triangle formed by two radii and the chord. If the radii make a central angle theta, dropping a perpendicular from the center to the chord bisects both the chord and the angle. Each half-chord has length r*sin(theta/2), so the full chord is 2r*sin(theta/2). The chord length increases with the central angle, reaching a maximum of 2r (the diameter) when theta = 180 degrees. For small angles, c is approximately r*theta (in radians), which is close to the arc length. The difference between chord length and arc length is negligible for small angles but becomes significant for larger ones. Historically, chord tables were precursors to trigonometric tables. Hipparchus (2nd century BCE) and Ptolemy (2nd century CE) compiled extensive chord tables in their astronomical works. Ptolemy's chord table, covering angles from 0.5 to 180 degrees in half-degree increments, was equivalent to a modern sine table and was used for astronomical calculations for over a millennium. In modern applications, chord length appears in: circular segment calculations (area of the region between a chord and the arc), gear design (tooth spacing on circular gears), satellite coverage (the ground footprint of a satellite's signal), and computer graphics (approximating circles with line segments). The sagitta (the height of the arc above the chord) is related to chord length by the formula: sagitta = r - sqrt(r^2 - (c/2)^2).

Formula Reference

Chord Length Formula

c = 2r * sin(theta/2)

Variables: r = radius, theta = central angle in radians

Worked Examples

Example 1: 60-Degree Chord

Find the chord length in a circle of radius 10 for a 60-degree central angle.

Step 1:r = 10, theta = 60 degrees = pi/3 radians
Step 2:c = 2 * 10 * sin(30 degrees) = 20 * 0.5 = 10

The chord length is 10 units (equal to the radius for a 60-degree angle).

Example 2: 90-Degree Chord

Find the chord length for a 90-degree central angle in a circle of radius 8.

Step 1:r = 8, theta = 90 degrees
Step 2:c = 2 * 8 * sin(45 degrees) = 16 * 0.7071 = 11.3137

The chord length is approximately 11.31 units.

Common Mistakes & Tips

  • !Forgetting to halve the angle before taking the sine. The formula uses sin(theta/2), not sin(theta).
  • !Using degrees directly in the sine function. If your calculator expects radians, convert first: radians = degrees * pi / 180.
  • !Confusing chord length with arc length. The chord is the straight-line shortcut; the arc is the curved path along the circle.

Related Concepts

Segment Area

The area of a circular segment (between a chord and its arc) uses both the chord length and the central angle.

Central Angle

The central angle determines the chord length and arc length for a given radius.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Central Angle Calculator

Circular Segment Area Calculator

Great Circle Distance Calculator

Frequently Asked Questions

When does the chord length equal the radius?

When the central angle is 60 degrees. In this case, the two radii and the chord form an equilateral triangle, so the chord length equals the radius.

What is the longest possible chord?

The diameter is the longest chord, corresponding to a central angle of 180 degrees. Its length is 2r, which matches c = 2r*sin(90) = 2r.

How do I find the central angle from a known chord length?

Rearrange the formula: theta = 2*arcsin(c/(2r)). This gives the central angle in radians; multiply by 180/pi to convert to degrees.