Arc Height Calculator (Radius + Arc Length Known)
Compute arc height (sagitta) precisely from a circle’s radius and arc length. Includes central angle, chord length, circular segment area, and a live visualization.
Inputs
Enter radius R and arc length s. Outputs will use the same length unit you choose.
Arc Height (Sagitta)
How to Calculate Arc Height When Radius and Arc Length Are Known
“Arc height” is a practical engineering phrase for the maximum rise of a circular arc above its straight chord. In classical geometry, this is the sagitta, typically denoted h. If you know the circle’s radius R and the arc length s along the curve, you can compute arc height accurately and consistently—without guessing the chord or the angle first. This is particularly valuable in fabrication (rolled plate, curved glass), civil layout (horizontal curves), mechanical design (pipe offsets), and any workflow where the arc length is easier to measure than the chord.
The key is to connect the linear arc length to the circle’s angular geometry. For a circle, arc length, radius, and central angle are tied together by a relationship so fundamental it effectively defines radian measure. Once you have the central angle, sagitta and chord fall out of right-triangle geometry with clean, stable formulas.
Core Definitions (Radius, Arc Length, Central Angle, Chord, Sagitta)
Before calculating, it helps to align on definitions:
- Radius (R): distance from circle center to any point on the circle.
- Arc length (s): length measured along the curved arc between two endpoints.
- Central angle (θ): angle (in radians) at the circle center subtending the arc.
- Chord length (c): straight-line distance between the arc endpoints.
- Arc height / sagitta (h): maximum distance from the chord to the arc (measured at the midpoint of the chord for a circular arc).
In many real-world contexts, “arc height” implicitly means the sagitta of the minor arc (the smaller arc, where 0 < θ ≤ π). If the arc length you have describes a major arc (larger than a semicircle), the same formulas can still be applied, but you should be explicit about interpretation because the geometry (and what you mean by “height”) can become ambiguous depending on which chord you choose and how the segment is oriented.
The Fundamental Relationship: θ = s / R (Why Radians Matter)
For a circle, arc length is proportional to the central angle:
θ = s / R (with θ in radians).
This is not just a convenient formula; it’s the defining property of radian measure. A radian is the angle that subtends an arc equal in length to the radius. So if your arc length equals your radius, θ = 1 radian. If your arc length is half the circumference, θ = π radians (a semicircle).
If your workflow uses degrees, convert only after you compute θ in radians:
θ(deg) = θ(rad) × 180 / π.
Arc Height (Sagitta) Formula Using Radius and Arc Length
Once θ is known, sagitta comes from the geometry of an isosceles triangle formed by two radii and the chord. Draw radii to the arc endpoints; split the triangle down the middle. The half-angle is θ/2 and the adjacent side equals R − h, while the hypotenuse equals R. That gives:
cos(θ/2) = (R − h) / R → h = R(1 − cos(θ/2)).
Substitute θ = s/R to compute sagitta directly from radius and arc length:
h = R(1 − cos(s / (2R))).
This is the cleanest “radius + arc length → arc height” expression. It’s unit-consistent: if R and s are both in meters, h will be in meters. If they’re in inches, h is in inches.
Chord Length (Helpful Companion Value)
Many downstream tasks need the chord length too (for example, laying out endpoints on a drawing or verifying fit in a jig). Using the same half-angle right triangle:
c = 2R sin(θ/2) = 2R sin(s / (2R)).
Segment Area (Optional, but Often Useful)
If you’re modeling a circular segment (e.g., fluid in a partially filled pipe or material cut from a disk), the segment area for the minor segment is:
A = (R²/2)(θ − sin θ), where θ is in radians.
Step-by-Step Calculation Workflow (Practical and Robust)
- Step 1: Validate inputs. Ensure R > 0 and s > 0. Confirm both are in the same unit system.
- Step 2: Compute θ. θ = s / R (radians).
- Step 3: Choose arc interpretation. In most fabrication and layout scenarios, you want the minor arc (θ ≤ π). If θ > π, confirm the measured arc is truly the minor arc; otherwise your “height” could represent an unintended geometry.
- Step 4: Compute sagitta. h = R(1 − cos(θ/2)).
- Step 5: Compute chord and checks. c = 2R sin(θ/2). Verify c ≤ 2R and 0 ≤ h ≤ R for minor arcs.
- Step 6: Sanity-check against limits. If s approaches 0, h should approach 0. If s approaches πR (semicircle), h should approach R.
Worked Example: Radius 10 m, Arc Length 8 m
Suppose R = 10 m and s = 8 m. First compute θ: θ = s/R = 8/10 = 0.8 rad. Then:
- h = 10(1 − cos(0.8/2)) = 10(1 − cos(0.4)) ≈ 10(1 − 0.92106099) ≈ 0.78939 m
- c = 2×10×sin(0.4) ≈ 20×0.38941834 ≈ 7.78837 m
Notice how the chord is slightly shorter than the arc (as expected), and the sagitta is modest because the central angle is less than 1 radian. This is a “gentle” curve.
Reference Table: Quick Intuition for Sagitta vs. Angle
The relationship between arc height and angle is nonlinear. Small increases in θ at larger angles create noticeably larger sagitta. The table below assumes a normalized radius R = 1 so you can scale results easily (multiply by your actual R).
| Central Angle θ (rad) | θ (deg) | h/R = 1 − cos(θ/2) | c/R = 2 sin(θ/2) |
|---|---|---|---|
| 0.2 | 11.459 | 0.004996 | 0.199667 |
| 0.5 | 28.648 | 0.031088 | 0.494808 |
| 1.0 | 57.296 | 0.122417 | 0.958851 |
| 1.5 | 85.944 | 0.268311 | 1.363278 |
| π | 180.000 | 1.000000 | 2.000000 |
Another Data Table: End-to-End Example Set (Same Radius, Different Arc Lengths)
If your radius is fixed (common in tooling or standardized bend radii), arc length becomes the main driver. The table below uses R = 100 (any unit) to show how outputs scale.
| R | s | θ = s/R (rad) | Chord c | Sagitta h |
|---|---|---|---|---|
| 100 | 10 | 0.100 | 9.996 | 0.125 |
| 100 | 50 | 0.500 | 49.481 | 3.109 |
| 100 | 100 | 1.000 | 95.885 | 12.242 |
| 100 | 200 | 2.000 | 168.294 | 45.970 |
Common Pitfalls (And How to Avoid Them)
1) Mixing units between R and s
Because θ = s/R, the two values must be in the same length unit. If R is in meters and s is in millimeters, θ will be wrong by a factor of 1000, and your sagitta will be nonsense. A reliable workflow is to pick one unit early (e.g., mm in manufacturing, m in civil) and keep everything consistent.
2) Using degrees inside trig functions
The formulas use radians. Most programming languages’ sin and cos functions also use radians. Always compute θ = s/R in radians, and only convert to degrees for display or documentation.
3) Confusing minor arc vs. major arc
If s is larger than πR, then θ is larger than π, which corresponds to more than a semicircle. In that range, “the arc” might be the major arc, but “arc height” might be intended for the complementary minor segment. If you’re physically measuring along a curve, double-check whether you followed the shorter path between endpoints or wrapped around the longer side of the circle.
4) Expecting sagitta to grow linearly
Sagitta grows roughly with the square of the angle for small angles. For small θ, a useful approximation is: h ≈ s² / (8R). This approximation comes from cos(x) ≈ 1 − x²/2 applied to h = R(1 − cos(s/(2R))). It’s fast for mental checks and helps you spot impossible results.
Where This Calculation Is Used (Real-World Applications)
- Metal rolling and bending: Given a roll radius and the developed arc length, sagitta tells you how “tall” the curve sits above the chord, which affects fit-up and clearance.
- Architecture and glazing: When specifying curved panels, the arc height affects support spacing, sealing strategies, and allowable tolerances.
- Civil engineering curves: While alignment design uses additional parameters, arc length and radius are foundational; sagitta can help visualize offsets.
- Machining and inspection: In metrology, sagitta relates to checking curvature using a straightedge and feeler gauge at midspan.
Authoritative References and Further Reading
For deeper context on circles, radians, and measurement systems, these references are helpful:
- National Institute of Standards and Technology (NIST) on SI units and measurement consistency: https://www.nist.gov/pml/weights-and-measures/si-units
- NASA Glenn educational overview of circle geometry concepts: https://www.grc.nasa.gov/WWW/k-12/airplane/circle.html
- MIT OpenCourseWare (calculus foundations that underpin radian measure and arc length ideas): https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/
Tip: If you’re documenting results for fabrication, record all three values—R, s, and computed h—plus the unit, because sagitta depends on both the radius and how much of the circle you used.