Circular Segment Chord Length, Arc & Sagitta Calculator
In precision machine shop geometry, a circular segment's radius from chord length C and sagitta height h is calculated as R = C² / (8h) + h / 2. Conversely, the sagitta arc height is h = R - √(R² - (C/2)²) and chord length is C = 2√(2Rh - h²) under ASME Geometric & Mechanical Standards (ASME Y14.5).
Circular Segment Chord Length, Arc Height (Sagitta) & Radius Calculator
Solve circular segment geometry from any two parameters: Radius R, Chord Length C, and Sagitta Height h (R = C²/(8h) + h/2, h = R - √(R² - (C/2)²)) under ISO 1101 & ASME Y14.5.
How to Calculate Arc Radius from Chord Length and Sagitta Height
When measuring large external rolls, shallow internal bores, or curved mold cavities on a surface plate, toolmakers frequently cannot reach the physical center of the circle with vernier calipers or height gages. Instead, they span a known reference chord length C across the arc using gage blocks or an optical comparator and measure the perpendicular sagitta rise h (the distance from the chord midpoint to the crest of the arc) using a dial indicator.
Applying the intersecting chords theorem from Machinery's Handbook (Industrial Press), the relationship states that the half-chord squared equals the product of the sagitta and the remaining diameter: (C/2)² = h × (2R - h). Expanding and isolating radius yields the exact mathematical formula R = C² / (8h) + h / 2. As verified in metrology calibrations at NIST (National Institute of Standards and Technology), this formula is purely deterministic and introduces zero geometric truncation error when computed with double-precision floating-point arithmetic.
| Nominal Radius R | Chord Length C | Sagitta / Arc Height h | Subtended Angle θ | Arc Length s | Segment Area A |
|---|---|---|---|---|---|
| 25.000 mm (0.9843") | 20.000 mm | 2.137 mm | 47.16° (0.823 rad) | 20.577 mm | 28.58 mm² |
| 25.000 mm (0.9843") | 35.355 mm | 7.322 mm | 90.00° (1.571 rad) | 39.270 mm | 178.10 mm² |
| 50.000 mm (1.9685") | 50.000 mm | 6.699 mm | 60.00° (1.047 rad) | 52.360 mm | 226.49 mm² |
| 50.000 mm (1.9685") | 70.711 mm | 14.645 mm | 90.00° (1.571 rad) | 78.540 mm | 712.39 mm² |
| 100.000 mm (3.9370") | 60.000 mm | 4.606 mm | 34.92° (0.609 rad) | 60.941 mm | 184.60 mm² |
| 100.000 mm (3.9370") | 100.000 mm | 13.397 mm | 60.00° (1.047 rad) | 104.720 mm | 905.97 mm² |
| 250.000 mm (9.8425") | 150.000 mm | 11.454 mm | 34.92° (0.609 rad) | 152.352 mm | 1,153.7 mm² |
Why Is Sagitta Measurement Critical for Bore and Flange Inspection?
Inspecting partial arcs—such as turbine casing segments, large pipe flanges, and crane wheels—presents a metrological challenge when less than 180° of the circumference exists. If an inspector measures a chord length of C = 60.000 mm and records a dial indicator sagitta deflection of h = 10.000 mm, our formula gives an exact radius of R = (3600 / 80) + 5 = 50.000 mm (Ø100.000 mm bore). The subtended angle is θ = 2 × arcsin(60 / 100) = 73.74° with an arc length of s = 64.350 mm.
Understanding the ratio h / C provides machinists with an immediate sensitivity check: for very shallow arcs where h / C < 0.05, tiny indicator errors of ±0.001 mm propagate into significant radius uncertainty because h appears in the denominator 8h. Machinists should maximize chord span across at least 60° of arc whenever feasible—see our CMM 3-Point Circle Center Calculator and Sine Bar Calculator.
| Height-to-Chord Ratio (h/C) | Subtended Angle θ (deg) | Radius Factor k_r (R/C) | Arc Length Factor k_s (s/C) | Area Factor k_a (A/C²) |
|---|---|---|---|---|
| 0.02 (Very Shallow Arc) | 9.16° | 6.2600 | 1.0011 | 0.0134 |
| 0.05 (Shallow Flare) | 22.84° | 2.5250 | 1.0066 | 0.0336 |
| 0.10 (Standard Machinist Crown) | 45.24° | 1.3000 | 1.0264 | 0.0677 |
| 0.134 (60° Segment) | 60.00° | 1.0000 | 1.0472 | 0.0906 |
| 0.20 (Deep Relief Arch) | 87.21° | 0.7250 | 1.1017 | 0.1375 |
| 0.293 (90° Quadrant Segment) | 90.00° | 0.7071 | 1.1107 | 0.1427 |
| 0.500 (Semicircle / Half Bore) | 180.00° | 0.5000 | 1.5708 | 0.3927 |
How to Convert Subtended Angle and Arc Length to Segment Area
In sheet metal stamping and CNC milling pocket clearing, calculating the cross-sectional area of a circular segment is necessary to determine total volumetric material removal and cutting forces. The net segment area equals the area of the circular sector minus the central triangular core: A = 0.5 × R² × (θ_{rad} - sin(θ_{rad})) under ISO International Metrology Standard (ISO 1101). Cross-reference with our Ball Nose Scallop Height Calculator and Bull-Nose End Mill Scallop Calculator.