14.142136
units
2.928932
units
15.707963
units
7.071068
units
28.539816
units²
14.142136
units
2.928932
units
15.707963
units
7.071068
units
28.539816
units²
The Chord Length Calculator determines the straight-line distance between two points on a circle, given the radius and the central angle that subtends the chord. The formula $$c = 2r\sin\left(\frac{\theta}{2}\right)$$ provides an exact result using basic trigonometry, and this calculator also computes the sagitta (the height of the arc above the chord).
A chord is any straight line segment whose endpoints lie on a circle. The diameter is a special chord that passes through the center and is the longest possible chord. All other chords are shorter than the diameter. Chords are fundamental objects in circle geometry and appear in Euclid's Elements as the basis for many propositions about circles.
Chord length calculations have numerous practical applications. In engineering, chords define the straight-line dimensions of curved structural elements. When a circular arch spans an opening, the chord length is the span width. In bridge design, the chord of a curved bridge deck determines the straight-line distance between abutments. In woodworking, when cutting a circular arc on a piece of material, the chord defines the straight edge of the cut.
In surveying and navigation, chord lengths are used to lay out circular curves on roads, railways, and canals. The surveyor places stakes along the curve, and the chord distance between consecutive stakes determines the practical spacing. Highway design standards specify curve geometry in terms of radius, central angle, and chord lengths.
The sagitta (also called the versine or arc height) is the perpendicular distance from the midpoint of the chord to the arc. It measures how much the arc 'bulges' outward from the chord. The sagitta is calculated as $$h = r(1 - \cos(\theta/2))$$. This measurement is essential in optics (lens curvature), architecture (arch height), and manufacturing (measuring circularity deviation).
The chord-angle relationship reveals interesting properties. For small angles, the chord length approximately equals the arc length (both approach $$r\theta$$). As the angle increases to 180°, the chord reaches its maximum value of $$2r$$ (the diameter). Beyond 180°, the chord length decreases again because the two endpoints begin moving closer together from the other side.
In music theory, the word 'chord' actually derives from a shortening of 'accord,' but the geometric concept of a chord has influenced the development of trigonometry. The ancient Greek astronomer Hipparchus created the first known table of chords around 150 BCE, which served the same purpose as modern sine tables. The relationship $$\text{chord}(\theta) = 2r\sin(\theta/2)$$ connects these historical chord tables directly to the sine function.
This calculator accepts radius and central angle in degrees, returning both the chord length and the sagitta for complete characterization of the chord-arc geometry.
The Chord Length Calculator uses the trigonometric chord formula:
$$c = 2r\sin\left(\frac{\theta}{2}\right)$$
Where:
Derivation: Drop a perpendicular from the center to the chord. This bisects both the chord and the central angle. In the resulting right triangle, the hypotenuse is $$r$$, the angle at the center is $$\theta/2$$, and the opposite side (half the chord) is $$r\sin(\theta/2)$$. The full chord is twice this: $$c = 2r\sin(\theta/2)$$.
Sagitta formula:
$$h = r - r\cos\left(\frac{\theta}{2}\right) = r\left(1 - \cos\frac{\theta}{2}\right)$$
The sagitta is the distance from the center of the circle to the chord, subtracted from the radius. Equivalently, it is the arc's maximum distance from the chord.
The Chord Length is the straight-line distance between the two points where the arc meets. It is always less than or equal to the diameter ($$2r$$), with equality only when the central angle is exactly 180°.
The Sagitta measures how high the arc rises above the chord. A sagitta of 0 means the chord coincides with the arc (degenerate case). A sagitta equal to the radius means the angle is 180° and the arc is a semicircle. The sagitta is useful for determining clearance heights under arches and for specifying the curvature of optical lenses.
Inputs
Results
θ = 90° = π/2. Chord = 2 × 10 × sin(45°) = 20 × 0.7071 = 14.14 units. Sagitta = 10 × (1 - cos(45°)) = 10 × 0.2929 = 2.93 units.
Inputs
Results
θ = 60°. Chord = 2 × 20 × sin(30°) = 40 × 0.5 = 20.00 units — the chord equals the radius! This creates an equilateral triangle when connected to the center. Sagitta = 20 × (1 - cos(30°)) ≈ 2.68 units.
The diameter is the longest chord, with length $$2r$$. It corresponds to a central angle of 180°. At this angle, $$c = 2r\sin(90°) = 2r$$.
The sagitta (from Latin for 'arrow') is the maximum distance from the chord to the arc. It is critical in optics for specifying lens curvature, in architecture for defining arch height, and in manufacturing for measuring circularity. The formula is $$h = r(1 - \cos(\theta/2))$$.
Rearrange the formula: $$\theta = 2\arcsin\left(\frac{c}{2r}\right)$$. For example, if $$c = 10$$ and $$r = 10$$, then $$\theta = 2\arcsin(0.5) = 2 \times 30° = 60°$$.
Beyond 180°, the two endpoints of the arc begin approaching each other from the far side of the circle. The chord still connects the endpoints by the shortest straight line, which gets shorter. At 360°, the chord length returns to 0 because both endpoints coincide.
For small angles, chord length ≈ arc length. As the angle increases, the arc (curved path) becomes progressively longer than the chord (straight path). The ratio chord/arc approaches 1 as $$\theta \to 0$$ and equals $$2/\pi \approx 0.637$$ at $$\theta = 180°$$.
Ancient Greek astronomers like Hipparchus and Ptolemy used tables of chord lengths instead of sine values. The relationship is $$\text{crd}(\theta) = 2R\sin(\theta/2)$$ in a circle of radius $$R$$. Indian mathematicians later developed the half-chord (jya), which evolved into our modern sine function.
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