A chord in geometry is a straight line segment whose endpoints both lie on the edge of a circle. Any two points you pick on a circle’s circumference can be connected to form a chord. The diameter, which passes through the center, is simply a special chord, and it’s the longest one possible.
How a Chord Differs From Other Circle Lines
Circles involve several types of lines that are easy to mix up. A chord is a line segment that starts and stops on the circle’s edge, staying inside the circle. A secant is a full line that passes through the circle and intersects the edge at two points, extending infinitely in both directions. A tangent touches the circle at exactly one point and never enters it. A radius connects the center to the edge, while a diameter connects two edge points through the center.
The key distinction is that a chord has two endpoints, both sitting on the circumference. It doesn’t need to pass through the center, and it doesn’t extend beyond the circle. If you stretched a chord outward in both directions so it kept going past the circle, it would become a secant.
The Diameter Is the Longest Chord
Euclid proved this in Book III of his Elements: of all chords in a circle, the diameter is the greatest, and chords closer to the center are always longer than those farther away. The reasoning is straightforward. A diameter equals twice the radius (2r). Any other chord forms a triangle with two radii drawn to its endpoints, and by the triangle inequality, that chord must be shorter than the sum of those two radii. Since the diameter equals exactly that sum, no other chord can match it.
This also works in reverse. The longest chord in a circle must pass through the center, making it a diameter by definition.
Calculating Chord Length
There are two common formulas for finding the length of a chord, depending on what information you have.
If you know the central angle: the chord length equals 2r × sin(θ/2), where r is the radius and θ is the central angle (the angle formed at the circle’s center by the two radii drawn to the chord’s endpoints). For example, in a circle with radius 10, a chord subtended by a 60° central angle has length 2 × 10 × sin(30°) = 10.
If you know the distance from the center: the chord length equals 2 × √(r² − d²), where r is the radius and d is the perpendicular distance from the chord to the center. This formula comes directly from the Pythagorean theorem. If you drop a perpendicular line from the center to the chord, it creates a right triangle with the radius as the hypotenuse. The closer the chord sits to the center, the longer it is, which confirms Euclid’s observation.
The Perpendicular Bisector Property
One of the most useful properties of chords: the perpendicular bisector of any chord passes through the center of the circle. In other words, if you find the midpoint of a chord and draw a line at a right angle to it, that line will hit the center. This works for every chord in the circle.
This property has a practical consequence. If you have a circular arc (say, a broken plate or a curved wall) and need to find the center of the full circle, you can draw two chords across the arc, construct their perpendicular bisectors, and the point where those bisectors cross is the center. Architects and engineers use this technique when working with curved structures.
The perpendicular from the center also bisects the chord’s corresponding arc. So if a perpendicular from the center cuts a chord in half, it splits the arc above it into two equal pieces as well.
Chords and Arcs
Every chord carves out an arc on the circle, and the relationship between them is consistent: equal chords intercept equal arcs within the same circle (or in circles with the same radius). A longer chord corresponds to a larger arc, and a shorter chord to a smaller one. This holds as long as you’re comparing chords in circles of the same size.
The converse is also true. If two arcs are equal, the chords connecting their endpoints are equal in length. This reciprocal relationship makes it possible to work backward from arc measurements to chord lengths, which is useful in problems involving circular geometry.
The Intersecting Chords Theorem
When two chords cross inside a circle, something elegant happens. If chord AC and chord BD intersect at point E, then the products of their segments are equal: AE × EC = DE × EB. This result dates back to Euclid’s Elements (Book III, Proposition 35).
As a concrete example, suppose two chords cross inside a circle. One chord is split into segments of 3 and 8 by the intersection point. The other chord is split into segments of 4 and some unknown length. Using the theorem: 3 × 8 = 4 × x, so x = 6. The unknown segment is 6 units long. This theorem shows up frequently in standardized math problems and in any situation where you need to find unknown lengths inside a circle.
Chords in Architecture and Design
Chords appear naturally in any structure that incorporates circular arcs. Arched windows, for instance, often use a chord as the flat bottom edge where the curved portion meets the rectangular frame below. The Gran Teatro Falla in Cadiz, Spain, built in 1905, features a row of arched windows where each circular top spans a 240° arc, with a chord connecting the endpoints of each arch to form the transition to the straight-edged window below.
Bridge design relies on chord geometry as well. The curved cables or arches of a bridge can be modeled as circular arcs, and the deck or structural supports act as chords. Knowing the relationship between a chord’s length, its distance from the center of curvature, and the radius lets engineers calculate load distribution and structural clearance without physically measuring every point along the curve.