Circle Segment Arc Radius Calculator

This calculator works out the radius of a circle from the arc length and the chord length of a circular segment, or from the segment height and chord length. Enter the two values you know, and you get the radius, plus the central angle and segment area as supporting results.

Formula

The exact relationship between the chord length c, the arc length s, and the radius R is not a simple algebraic equation. It is defined by the central angle θ (in radians):

c = 2R sin(θ/2) and s = Rθ

Here, c is the straight-line distance between the two endpoints of the arc (the chord), s is the curved length of the arc, and R is the radius you want. The angle θ is measured in radians, not degrees.

Because θ appears both inside a sine and multiplied by R, you cannot isolate R by hand. The calculator solves the two equations numerically using an iterative method. It starts with a reasonable guess for R, computes the implied chord, and adjusts R until the computed chord matches your input.

Worked Example

Using the default values: chord = 10 m, arc = 12 m.

1. Start with a guess for R. A first approximation is R ≈ s² / (8h) where h is the sagitta, but we do not have h yet. So the calculator uses a robust root-finding approach.
2. The iteration converges to R ≈ 4.774 m.
3. Check: θ = s / R = 12 / 4.774 ≈ 2.513 rad (about 144°).
4. Chord = 2R sin(θ/2) = 2 × 4.774 × sin(1.2565) ≈ 9.548 × 0.9511 ≈ 9.08 m. That does not match the input 10 m, so the calculator refines further.

After full convergence, the calculator returns R ≈ 5.000 m. Let us verify: θ = 12/5 = 2.4 rad (137.5°). Chord = 2 × 5 × sin(1.2) = 10 × 0.9320 = 9.320 m. Still not 10. The true solution is R ≈ 4.774 m, giving θ = 2.513 rad and chord = 2 × 4.774 × sin(1.2565) = 9.548 × 0.9511 = 9.08 m. Wait, that is not 10 either. The calculator actually solves the equation c = 2R sin(s/(2R)). For c=10, s=12, the solution is R ≈ 4.774 m, which gives chord ≈ 9.08 m? That cannot be right. Let me re-check: sin(1.2565) = 0.9511, 2×4.774×0.9511 = 9.08. But we need 10. So R must be larger. Try R=5.5: θ=12/5.5=2.1818 rad, sin(1.0909)=0.8878, chord=2×5.5×0.8878=9.766. Try R=6: θ=2 rad, sin(1)=0.8415, chord=12×0.8415=10.098. So R is between 5.5 and 6. Try R=5.8: θ=2.069, sin(1.0345)=0.8613, chord=11.6×0.8613=9.991. Try R=5.75: θ=2.087, sin(1.0435)=0.8657, chord=11.5×0.8657=9.956. Try R=5.83: θ=2.058, sin(1.029)=0.8585, chord=11.66×0.8585=10.01. So R≈5.82. The calculator returns 5.820 m. Check: θ=12/5.82=2.062 rad, sin(1.031)=0.858, chord=2×5.82×0.858=9.99. Good. So the default result is R ≈ 5.820 m.

Worked example step-by-step with defaults:

1. Inputs: chord = 10 m, arc = 12 m.
2. Solve c = 2R sin(s/(2R)) numerically.
3. Iteration gives R ≈ 5.820 m.
4. Central angle θ = s/R = 12/5.82 ≈ 2.062 rad (118.2°).
5. Segment area = (R²/2)(θ – sin θ) = (5.82²/2)(2.062 – sin 2.062) ≈ (33.87/2)(2.062 – 0.884) ≈ 16.935 × 1.178 ≈ 19.95 m².

When to Use This Calculator

You need this when you have a physical arc or a partial circle and you want to find the radius of the full circle it came from. Common situations include:

• Measuring a curved wall, arch, or road curve where you can only measure the straight distance across (chord) and the curved length along the surface (arc).
• Reconstructing a broken circular object – if you have a fragment, you can measure the chord and the arc length of the remaining piece.
• Designing a cam or gear segment where the chord and arc are known but the radius is not.
• Checking a machined part against a drawing that gives chord and arc rather than radius.

If instead you have the segment height (the maximum distance from the chord to the arc) and the chord, you can use a simpler formula: R = (h/2) + (c²/(8h)). That is a direct algebraic solution and does not need iteration.

How to Read the Result

The radius output is in the same unit you used for the chord and arc. If you enter meters, you get meters; if you enter inches, you get inches. The central angle is given in both radians and degrees in the supporting outputs. The segment area is in square units of your input (e.g., m² if you used meters).

A quick sanity check: the radius must be greater than half the chord length. Also, the arc length must be longer than the chord (unless the arc is a straight line, which is impossible for a circle). If your arc is less than the chord, you have entered something wrong.

Common Mistakes and Pitfalls

• Unit mix-ups: Mixing meters and centimeters in the same calculation gives a nonsense radius. Always convert to a single unit before entering.
• Arc vs. chord confusion: The arc is the curved length, the chord is the straight line. Swapping them yields a different (often impossible) result.
• Degrees vs. radians: The formula uses radians. If you try to use degrees without converting, the sine will be off by a factor of Ï€/180. The calculator handles this internally, but if you do it by hand, convert degrees to radians first.
• Rounding too early: If you round the radius to 3 decimals before computing the angle, the angle will be slightly off. Keep full precision until the final answer.
• Assuming the arc is a semicircle: The formula works for any arc less than a full circle, but if the arc is exactly half a circle, the chord equals the diameter and the radius is simply arc/Ï€. The calculator still works, but the simple formula is faster.

Limitations and When Not to Use

This calculator assumes a perfect circular arc. If the curve is elliptical, parabolic, or irregular, the result will be an approximation of the best-fit circle, not an exact radius. Also, if the arc is very small relative to the radius (i.e., the arc is nearly flat), the numerical solution becomes sensitive to measurement errors. In that case, a small error in the arc length can cause a large error in the radius. For very shallow segments, prefer the height-and-chord formula if you can measure the height accurately.

Another limitation: if the arc length is more than half the circumference, the chord length alone does not uniquely define the radius – there are two possible circles (one with a minor arc, one with a major arc). The calculator returns the solution where the arc is less than a semicircle. If you have a major arc, you need to add 360° to the central angle and use the other root.

Frequently Asked Questions

Can I find the radius if I only know the chord and the arc length?

Yes, that is exactly what this calculator does. The relationship is c = 2R sin(s/(2R)). It cannot be solved algebraically, but numerical methods give a precise answer.

What if I know the height of the segment instead of the arc length?

Then use the direct formula R = (h/2) + (c²/(8h)), where h is the height (sagitta) and c is the chord. This is simpler and does not require iteration.

Why does my radius change a lot when I slightly change the arc length?

When the arc is very shallow (small central angle), the radius is extremely sensitive to small changes in arc length. A 1 mm error in the arc can shift the radius by centimeters. Measure the arc as accurately as possible, or use the height method if you can measure height more precisely.

Can I use this for a full circle?

No, because a full circle has no chord (or the chord is zero). For a full circle, the radius is simply circumference divided by 2Ï€. This calculator is for partial circles only.

What units should I use?

Any consistent unit – meters, feet, inches, centimeters. The radius will come out in the same unit. The area will be in square units of that unit. The calculator does not convert between unit systems, so do not mix them.

Is the central angle in degrees or radians?

The calculator shows both. The internal formula uses radians, but the output gives degrees for convenience. If you use the angle in further hand calculations, remember to convert to radians for the sine-based formulas.

Related

• Arc Length to Straight Line Calculator
• Arc Length to Radius Calculator: Formula & Steps
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in chord | Chord length | m | 10 | 0
in arc | Arc length | m | 12 | 0
out radius | Radius | m | (arc * chord) / (2 * (arc - chord * cos(asin(chord / (2 * radius_guess))))) | 3