Chord Length Calculator
Calculate the chord length of a circle from radius and central angle. Free online chord length geometry calculator with formula, step-by-step results, and instant answers for students and professionals.
About This Calculator
What Is a Chord of a Circle?
A chord is a straight-line segment whose endpoints both lie on the circumference of a circle. Unlike the radius or diameter, a chord does not necessarily pass through the center. The longest chord in any circle is the diameter, which passes through the center and equals twice the radius. Every other chord is shorter than the diameter.
Chord Length Formula
The chord length is calculated using the formula L = 2r sin(theta/2), where L is the chord length, r is the radius of the circle, and theta is the central angle (in radians) that subtends the chord. When working in degrees, convert the angle to radians first by multiplying by pi/180. This formula is derived from the law of cosines and the geometry of an isosceles triangle formed by two radii and the chord.
Properties of Chords
Chords have several important geometric properties: (1) Equal chords subtend equal central angles, and vice versa. (2) A perpendicular from the center to a chord bisects the chord. (3) Chords equidistant from the center have equal length. (4) The chord length increases as the central angle increases from 0 deg to 180 deg, then decreases symmetrically from 180 deg to 360 deg.
Real-World Applications
Chord length calculations are essential in structural engineering (designing arched bridges, roof trusses, and curved beams), architecture (arches, domes, and circular windows), woodworking (cutting circular segments), astronomy (measuring angular distances between celestial bodies), and surveying (calculating straight-line distances across curved terrain).
How To Use This Calculator
Enter the radius of the circle in any unit of length (meters, feet, inches, centimeters) and the central angle in degrees. Click Calculate to instantly compute the chord length. The radius and angle are saved to the page URL, allowing you to bookmark or share your calculation with others.
Relationship Between Chord, Arc, and Sector
For any given central angle, the chord, arc, and sector are three related geometric features: the chord is the straight line connecting the endpoints, the arc is the curved portion of the circumference between them, and the sector is the pie-shaped region bounded by the two radii and the arc. This calculator focuses on the chord length, while our related arc length and sector area tools handle the other two.
Frequently Asked Questions
What is the chord length formula?
The chord length formula is L = 2r sin(theta/2), where r is the radius of the circle and theta is the central angle subtended by the chord. When the angle is given in degrees, first convert to radians by multiplying by pi/180 before applying the formula.
How do you calculate chord length from radius and angle?
Enter the radius and central angle in degrees into the calculator. The tool automatically converts the angle to radians and applies the formula chord = 2 x radius x sin(angle/2). Results are displayed instantly with two decimal place precision.
What is the difference between chord length and arc length?
A chord is the straight-line distance between two points on a circle's circumference. Arc length is the curved distance along the circle between the same two points. The arc is always longer than the chord for any angle between 0 deg and 360 deg, and both become equal only at 0 deg.
What units does the chord length calculator use?
The calculator accepts any unit of length for the radius (e.g., meters, feet, inches, centimeters). The resulting chord length will be in the same unit as the radius. The central angle is always entered in degrees and is automatically converted to radians for the calculation.
What is the longest possible chord in a circle?
The longest possible chord in a circle is the diameter, which occurs when the central angle is 180 deg. The diameter chord length equals 2r (twice the radius). For any angle less than 180 deg, the chord is shorter than the diameter; for angles greater than 180 deg, the chord length decreases again.
Can the central angle be greater than 180 degrees?
Yes, central angles from 180 deg to 360 deg are valid. As the angle increases beyond 180 deg, the chord length decreases symmetrically. For example, a 270 deg angle produces the same chord length as a 90 deg angle, since sin(270 deg/2) = sin(135 deg) = sin(45 deg). The chord reaches its maximum at 180 deg.