Circle Chord Length Calculator
A chord is a straight line segment connecting two points on the circumference of a circle. Every chord divides the circle into two arcs — a shorter minor arc and a longer major arc. The longest chord of any circle passes through its center and is called the diameter. Understanding chords is fundamental to plane geometry and has applications in architecture, structural engineering, road design, and academic mathematics.
Two Ways to Calculate Chord Length
This calculator supports two distinct methods for finding chord length, each suited to different available data.
Mode A — Central Angle and Radius
When you know the radius r and the central angle θ subtended by the chord at the center of the circle, use:
L = 2 × r × sin(θ / 2)
The central angle must be entered in either degrees (0° to 360°) or radians (0 to 2π). For example, a radius of 10 m and a central angle of 60° gives a chord length of exactly 10 m, because 2 × 10 × sin(30°) = 2 × 10 × 0.5 = 10. When the angle is 180°, the chord equals the diameter (2r).
Mode B — Distance from Center and Radius
When you know the perpendicular distance d from the center of the circle to the chord, use:
L = 2 × √(r² − d²)
This formula follows directly from the Pythagorean theorem applied to the right triangle formed by the radius, the perpendicular from the center to the chord, and the half-chord. The distance d must satisfy 0 ≤ d < r. When d = 0 the chord passes through the center and equals the diameter. When d approaches r the chord length approaches zero (a degenerate chord at the edge).
Secondary Outputs Explained
Beyond the chord length itself, the calculator provides several related values:
- Half Chord Length — half of L, useful in construction layout and site surveying where only one side of a symmetric layout is measured.
- Sagitta (h) — the perpendicular height from the midpoint of the chord up to the arc. Formula:
h = r − √(r² − (L/2)²). The sagitta tells you how much the arc "bows" above the chord and is critical in arch engineering, lens grinding, and shipbuilding. - Arc Length (s) — the length of the arc that subtends the chord:
s = r × θ(θ in radians). Arc length is always longer than chord length for any arc less than a full circle. - Central Angle — shown when using Mode B (distance from center), this is the angle derived from
θ = 2 × arcsin(L / 2r). It describes how wide the chord is from the circle's center perspective. - Is Diameter? — a flag set to "Yes" when the chord length equals 2r within floating-point tolerance.
Unit Support and Precision
All length inputs and outputs use the same selected unit — millimetres, centimetres, metres, kilometres, inches, feet, or miles. Internally, the calculator converts to metres for computation and back for display. Angles can be entered in degrees or radians, and output precision is adjustable from 0 to 10 decimal places.
Real-World Applications
Chord length calculations arise in many practical fields. In civil engineering, road designers use chords to approximate curve lengths between survey stations. In architecture, the chord and sagitta define the geometry of arches, barrel vaults, and curved facades. In manufacturing, chord measurements help machinists verify the diameter of turned components using a height gauge without direct access to the center. In astronomy, the chord length of a transit across a stellar disk can be used to determine the size of planets or moons.
Interactive Diagram
The calculator includes an SVG diagram that updates dynamically as you change inputs. It shows the circle, the chord labeled L, the radius line labeled r, the central angle arc labeled θ, the perpendicular distance line labeled d, and the sagitta labeled h. This visual representation helps you confirm that the inputs match the geometric scenario you have in mind before reading the numeric outputs.
Step-by-Step Solution
A collapsible formula panel shows each substitution step — from the raw formula through intermediate values to the final result. This is especially helpful for students learning plane geometry who need to understand not just the answer but the reasoning process. The steps are presented in the same format as a worked example in a textbook, making it easy to follow along or cite in homework.