A chord is the straight-line segment connecting any two points on a circle's circumference. Despite its simple definition, chord length is a critical measurement woven into structural engineering, CNC machining, architectural design, and dozens of applied mathematics disciplines.
Manually computing chord length—along with its dependent quantities like the sagitta, apothem, and circular segment area—is tedious and error-prone. An automated approach eliminates unit-conversion mistakes and enforces geometric boundary conditions, ensuring every derived value is internally consistent before it reaches a blueprint or a machining program.
Required Project Parameters
Before performing any chord-related computation, the following geometric variables must be established. Typically, only two are needed (one of which is usually the radius) to derive all remaining values:
- Radius ($r$): The distance from the center of the circle to any point on its circumference. This is the fundamental geometric constraint governing every other measurement.
- Central Angle ($\theta$): The angle subtended by the chord at the center of the circle, expressed in degrees or radians.
- Apothem ($d$): The shortest perpendicular distance from the circle's center to the chord. It represents the "depth" of the chord within the circle's interior.
- Sagitta ($h$): Also called the arc height, the perpendicular distance from the midpoint of the chord to the arc. Essential for arch design and curvature analysis.
- Arc Length ($s$): The actual curvilinear distance measured along the circle between the chord's two endpoints, as opposed to the straight-line chord distance.
Trigonometric Derivation of Chord and Its Dependent Quantities
The Primary Chord Length Formula
The standard trigonometric relationship for computing chord length $c$ from a known radius $r$ and central angle $\theta$ (in radians) is:
$$c = 2r \sin\left(\frac{\theta}{2}\right)$$
This formula arises from bisecting the central angle. Drawing a line from the center of the circle perpendicular to the chord creates two congruent right triangles. Each triangle has a hypotenuse of $r$ and an opposite side of $\frac{c}{2}$, yielding the relationship $\sin!\left(\frac{\theta}{2}\right) = \frac{c/2}{r}$.
Degree-to-radian conversion is mandatory before applying any trigonometric function:
$$\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$$
Apothem and Its Pythagorean Verification
The apothem $d$ is derived directly from the cosine of the half-angle:
$$d = r \cos\left(\frac{\theta}{2}\right)$$
An equivalent Pythagorean derivation provides an independent cross-check, which is invaluable when working with physical materials like steel beams or CNC-cut panels:
$$d = \sqrt{r^2 - \left(\frac{c}{2}\right)^2}$$
Both expressions must yield identical results. Any discrepancy signals an input or rounding error.
Sagitta: The Collinear Complement of the Apothem
The sagitta $h$ and the apothem $d$ are collinear segments along the perpendicular bisector of the chord. Together, they compose the full radius:
$$d + h = r$$
Therefore:
$$h = r - d = r - r\cos\left(\frac{\theta}{2}\right) = r\left[1 - \cos!\left(\frac{\theta}{2}\right)\right]$$
This identity, $d + h = r$, serves as an immediate self-check for accuracy in any physical fabrication context. If the measured sagitta plus the measured apothem does not equal the known radius, the cut or bend is dimensionally incorrect.
Boundary condition: The sagitta cannot exceed the diameter ($h \leq 2r$), and the apothem cannot exceed the radius ($d \leq r$).
Arc Length Computation
The arc length $s$ subtended by the central angle is a purely linear function of $\theta$:
$$s = r\theta$$
where $\theta$ is strictly in radians. This value is always greater than or equal to the chord length $c$, with equality occurring only in the degenerate case where both approach zero.
Circular Segment, Sector, and Triangle Areas
Three distinct area quantities emerge from a chord and its associated arc:
Sector Area (the "pie slice"):
$$A_{\text{sector}} = \frac{1}{2} r^2 \theta$$
Triangle Area (formed by the two radii and the chord):
$$A_{\text{triangle}} = \frac{1}{2} r^2 \sin(\theta)$$
Segment Area (the region between the chord and the arc):
$$A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{1}{2} r^2 \left(\theta - \sin\theta\right)$$
These formulas handle any central angle from $0$ to $2\pi$ consistently, including angles exceeding $\pi$ (180°), which define major segments.
Segment Perimeter
The perimeter of the circular segment is the sum of the chord and the arc:
$$P_{\text{segment}} = c + s = 2r\sin\left(\frac{\theta}{2}\right) + r\theta$$
Reference Tables for Standard Chord Geometries
Chord Ratios for Common Central Angles ($r = 1$)
| Central Angle ($\theta$) | Chord Length ($c$) | Apothem ($d$) | Sagitta ($h$) | Arc Length ($s$) |
|---|---|---|---|---|
| 30° | 0.5176 | 0.9659 | 0.0341 | 0.5236 |
| 45° | 0.7654 | 0.9239 | 0.0761 | 0.7854 |
| 60° | 1.0000 | 0.8660 | 0.1340 | 1.0472 |
| 90° | 1.4142 | 0.7071 | 0.2929 | 1.5708 |
| 120° | 1.7321 | 0.5000 | 0.5000 | 2.0944 |
| 150° | 1.9319 | 0.2588 | 0.7412 | 2.6180 |
| 180° | 2.0000 | 0.0000 | 1.0000 | 3.1416 |
Area Components for Common Central Angles ($r = 1$)
| Central Angle ($\theta$) | Sector Area | Triangle Area | Segment Area | Segment Perimeter |
|---|---|---|---|---|
| 30° | 0.2618 | 0.2500 | 0.0118 | 1.0412 |
| 45° | 0.3927 | 0.3536 | 0.0391 | 1.5508 |
| 60° | 0.5236 | 0.4330 | 0.0906 | 2.0472 |
| 90° | 0.7854 | 0.5000 | 0.2854 | 2.9850 |
| 120° | 1.0472 | 0.4330 | 0.6142 | 3.8265 |
| 150° | 1.3090 | 0.2500 | 1.0590 | 4.5499 |
| 180° | 1.5708 | 0.0000 | 1.5708 | 5.1416 |
Chordal Deviation vs. Segment Count for CNC Approximation ($r = 100$ mm)
When approximating a full circle with $n$ linear segments (chords), the chordal deviation $\delta$ (maximum perpendicular error between the chord and the true arc) equals the sagitta of each segment:
$$\delta = r\left[1 - \cos!\left(\frac{\pi}{n}\right)\right]$$
| Number of Segments ($n$) | Chord Length (mm) | Arc per Segment (mm) | Chordal Deviation $\delta$ (mm) | Surface Finish Impact |
|---|---|---|---|---|
| 12 | 51.76 | 52.36 | 1.340 | Visibly faceted |
| 24 | 26.11 | 26.18 | 0.341 | Noticeable under inspection |
| 48 | 13.08 | 13.09 | 0.086 | Acceptable for rough machining |
| 96 | 6.54 | 6.54 | 0.021 | Fine finish threshold |
| 360 | 1.745 | 1.745 | 0.0015 | Near-optical smoothness |
Applied Engineering Interpretation of Chord Geometry
Structural Arch Design and Sagitta-Dependent Thrust
In structural engineering, the sagitta directly governs the mechanical behavior of arched members. A shallow sagitta (small $h$ relative to $c$) produces a nearly flat arch, resulting in significantly higher horizontal thrust at the supports.
Conversely, a deep arch with a large sagitta distributes load more vertically, reducing the horizontal force component. Quantifying the sagitta with precision is therefore not merely a geometric exercise—it determines whether abutments require additional reinforcement against lateral displacement.
The ratio $\frac{h}{c}$ is commonly called the rise-to-span ratio and is a primary classification parameter in arch bridge design.
CNC Machining: Chordal Deviation as Surface Quality Metric
The distinction between chord and arc is perhaps nowhere more consequential than in CNC machining. Most CNC controllers approximate curves as a series of linear toolpath segments—effectively, chords.
The perpendicular discrepancy between each chord and the true arc is the chordal deviation (also termed sagitta error). This deviation is the primary source of surface finish inaccuracy on curved workpieces.
A machinist targeting a surface roughness of $R_a \leq 0.8 \,\mu\text{m}$ must ensure the chordal deviation is well below this threshold. From the formula $\delta = r[1 - \cos(\pi/n)]$, increasing the segment count $n$ reduces $\delta$ quadratically, but at the cost of longer program files and slower processing speed.
How the Central Angle Governs All Derived Quantities
The central angle $\theta$ is the master variable in chord geometry. Every other quantity—chord length, apothem, sagitta, arc length, and all three areas—is a direct or indirect function of $\theta$ and $r$.
- Increasing $\theta$ from 0° toward 180° causes chord length to increase monotonically toward the diameter $2r$, while the apothem decreases toward zero and the sagitta increases toward $r$.
- At $\theta = 60°$, the chord equals the radius ($c = r$), forming an equilateral triangle with the two radii. This is the geometric basis of the regular hexagon.
- At $\theta = 180°$, the chord becomes the diameter, the apothem vanishes, and the segment area equals a semicircle.
- For $\theta > 180°$, the calculation produces a major segment, where the segment area exceeds the semicircular area. The formulas remain algebraically consistent; no special-case logic is required.
The $d + h = r$ Integrity Check in Fabrication
For anyone cutting, bending, or forming material to a circular profile, the identity $d + h = r$ provides a zero-cost dimensional verification. After fabrication:
- Measure the chord length $c$ directly.
- Measure the sagitta $h$ with a depth gauge at the chord's midpoint.
- Compute $d = \sqrt{r^2 - (c/2)^2}$.
- Verify that $d + h = r$ within acceptable tolerance.
A failure in this check indicates either the radius of curvature drifted during forming or the chord endpoints are mislocated.
Frequently Asked Questions
Start from the identity $h = r - d$, which gives $d = r - h$. Since $d = \sqrt{r^2 - (c/2)^2}$, substitute and solve for $c$:
$$(r - h)^2 = r^2 - \left(\frac{c}{2}\right)^2$$
Expanding and simplifying:
$$c = 2\sqrt{2rh - h^2}$$
This is one of the most practical formulas in field measurement. Surveyors and machinists frequently know $r$ and can easily measure $h$ with a straightedge and depth gauge, making this the preferred derivation when the central angle is not directly available.
Chord length determines the size of each linear toolpath segment, but it is the chordal deviation $\delta$ that directly manifests as surface error on the finished part. Two toolpaths can have identical chord lengths yet produce different chordal deviations if applied to arcs of different radii.
The deviation $\delta = r[1 - \cos(\theta/2)]$ shows that for a fixed chord count, larger radii produce larger absolute deviations. This is counterintuitive—larger, gentler curves actually require more segments to maintain the same surface finish tolerance. Modern CAM software allows users to specify a maximum chordal tolerance (e.g., 0.01 mm), then automatically calculates the required segment density.
The formula $A_{\text{segment}} = \frac{1}{2}r^2(\theta - \sin\theta)$ remains mathematically valid for all $\theta \in (0, 2\pi)$. When $\theta > \pi$, the $\sin\theta$ term becomes negative, which means $(\theta - \sin\theta)$ increases beyond the semicircular value.
The result correctly computes the major segment area—the larger region on the opposite side of the chord from the minor arc. Legacy drafting practices sometimes required manually switching between minor and major segment formulas, but the trigonometric expression handles this transition seamlessly. At $\theta = 2\pi$, the segment area equals the full circle area $\pi r^2$, confirming algebraic consistency.
Precision Through Automated Geometric Computation
Chord geometry sits at the intersection of pure mathematics and physical fabrication. A single misapplied formula or a forgotten degree-to-radian conversion can cascade into costly structural errors, rejected machined parts, or unsafe arch designs.
Automated computation enforces unit consistency, applies boundary constraints (sagitta $\leq 2r$, apothem $\leq r$), and delivers all dependent quantities—segment area, sector area, triangle area, segment perimeter—simultaneously from a minimal set of known values. The built-in geometric identity $d + h = r$ provides an intrinsic verification layer that manual calculation typically omits.
For engineers, architects, machinists, and surveyors, replacing error-prone hand calculations with validated mathematical estimation is not a convenience—it is a professional standard of care.