Exact answer
1 arc second to degree is 1/3600 degree, or 0.000277777778 degree. 1 arc second in radians is pi/648000 rad, or about 4.848136811e-6 rad.
Hybrid Tool + Report
1 Arc Second in Radians: Degree, Radian, and Radius Tool
Arc Second to Degree and Radian Converter
Tool LayerStart with 1 arc second in radians, convert to degree and radian units, then project the angle to circular arc length at your working radius. Default answer: 0.000277777778 deg = 4.848136811e-6 rad.
Empty state: enter an angle and press Convert. The result will show arc-seconds, degrees, radians, micro-radians, arc-length projection, and a next action.
Ask About Arc-Second ToleranceBaseline: 1 arc second = 1/3600 degree = pi/648000 rad = 4.848136811e-6 rad. Circular arc-length projection uses
shift um = radians x radius mm x 1000.Default degree
0.000277777778 deg
Default radians
4.848136811e-6 rad
At 100 mm
0.484814 um
Quick presets
Use this single canonical page to answer arc second to degree and 1 arc second in radians. Convert first, then use the evidence and boundary sections to decide whether the number is meaningful for a rotary-axis tolerance.
Published: 2026-06-17 | Last reviewed: 2026-06-17
1 arc-sec in degree
0.000277777778
1 arc-sec in radians
4.848136811e-6
At 100 mm radius
0.484814 um
Full revolution
1,296,000 arc-sec
Direct Answer
1 Arc Second in Radians and Degrees
1 arc second equals 0.000277777778 degree and 4.848136811095e-6 rad. This page treats "1 arc second in radians" as an alias of the canonical arc-second-to-degree conversion because the formula chain is identical.
Exact degree
1 / 3600 deg
Exact radian
pi / 648000 rad
Micro-radian
4.848136811 urad
Executive Summary
Tool + reportSingle intent cluster
The query "1 arc second in radians" is the same conversion task as arc second to degree, so this page keeps both answers on one canonical URL.
Radian value drives real checks
At a 100 mm radius, 1 arc second is about 0.484814 um of circular arc length. The same angle at 500 mm is about 2.424 um.
Conversion is not capability
The math is exact, but rotary-axis acceptance still depends on backlash, encoder error, thermal drift, and measurement uncertainty.
Use the tool first
For RFQs and tolerance translation, convert to radians first, then compare the projected linear shift with your actual tolerance limit.
Use this page when
- Engineers translating arc-second specifications into degrees, radians, or linear displacement.
- Buyers checking whether a rotary-axis tolerance request is physically meaningful at the working radius.
- Content or documentation teams that need one authoritative answer for arc second to degree and 1 arc second in radians.
Do not use it as final proof when
- Teams trying to prove machine accuracy from a unit conversion alone.
- Fixture or Abbe-offset studies where chord displacement, cosine error, or full transformation math is required.
- Final acceptance work without measured backlash, encoder, repeatability, and thermal data.
Decision signal map
The page flow is intentionally tool-first: calculate the value, interpret the tolerance ratio, then read only the report sections needed for confidence.
Conversion Table
Exact identityValues are derived from the NIST angle identity. Keep the source formula beside converted values when moving the number into RFQ, CAD, spreadsheet, or motion-control notes.
| Input | Degree | Radian | Micro-radian | Linear at 100 mm (um) | Use |
|---|---|---|---|---|---|
| 1 arc second | 0.000277777778 | 4.848136811095e-6 | 4.848136811 | 0.484814 | Default direct answer and alias query target. |
| 10 arc seconds | 0.002777777778 | 4.848136811095e-5 | 48.481368111 | 4.848137 | Common low arc-second class screening value. |
| 60 arc seconds | 0.016666666667 | 2.908882086657e-4 | 290.888208666 | 29.088821 | Equivalent to 1 arc-minute; useful for avoiding unit confusion. |
Arc-Length Projection
Why Radians Matter After Conversion
Radians turn angle into length. The same 1 arc second value gives a circular arc length of about 0.484814 um at 100 mm, about 1.454441 um at 300 mm, and about 2.424068 um at 500 mm.
100 mm radius
0.484814 um
300 mm radius
1.454441 um
500 mm radius
2.424068 um
Chord delta at 100 mm
< 0.000001 um
Projection Model Boundaries
Arc vs chordThe calculator reports circular arc length because the radian is defined through arc length and radius. Use the table below to decide when that is sufficient and when a different geometry model is required.
| Model | Formula | Best use | Limit / counterexample | Evidence |
|---|---|---|---|---|
| Circular arc length | s = theta x r | Translating a small angular error into path length at a known rotary radius. | Only describes motion along the circular arc. It does not include fixture offsets, straight-line endpoint displacement, or machine compliance. | NIST SP 330 Section 5 NIST SP 330 Section 5 defines plane angle as theta = s/r rad. |
| Chord displacement | c = 2r x sin(theta / 2) | Straight-line endpoint checks, optical comparator geometry, or when the measured feature spans endpoints rather than arc path. | For 1 arc second at 100 mm, chord and arc differ by less than 0.000001 um; this remains a geometry choice, not a capability proof. | Derived from geometry Derived trigonometry; page tool reports chord delta as a boundary note. |
| Installed fixture transform | Site-specific kinematic model | Offset payloads, stacked axes, probes away from the centerline, or acceptance work with Abbe error risk. | No reliable public universal coefficient exists for a specific machine/fixture/load combination. | No reliable public data for a specific installation Pending confirmation: requires measured machine geometry and acceptance procedure. |
Evidence and Source Boundaries
Reviewed 2026-06-17The unit conversion is source-backed; the engineering interpretation is explicitly bounded. Rows marked "derived" are formulas from the stated identity, not independent measurement data.
| Topic | Evidence | Boundary | Source | Date |
|---|---|---|---|---|
| Angle identity | NIST SP 330 lists degree as pi/180 rad, minute as pi/10800 rad, and second of plane angle as pi/648000 rad. | This proves the conversion identity only; it does not validate any specific rotary axis. | NIST SP 330 Section 4 | 2019 SI edition; reviewed 2026-06-17 |
| Radian-to-radius model | NIST SP 330 defines plane angle in radians as circular arc length divided by radius, theta = s/r rad. | The radius projection is exact for circular arc length; it becomes an approximation only when used as chord or full fixture displacement. | NIST SP 330 Section 5 | 2019 SI edition; reviewed 2026-06-17 |
| Guide to SI usage | NIST SP 811 Chapter 5 repeats that second of plane angle is 1/60 minute and pi/648000 rad. | SP 811 is a usage guide and has not been updated for the 2019 SI revision; use SP 330 as the current primary SI source. | NIST SP 811 Chapter 5 | 2008 edition; reviewed 2026-06-17 |
| Alias query answer | 1 arc second in radians = 4.848136811e-6 rad, matching pi/648000 rad. | Keep the answer on this canonical arc-second-to-degree page to avoid duplicate near-identical pages. | Derived from NIST angle identity | Model reviewed 2026-06-17 |
| Linear projection | Arc length is angle in radians times radius. At 100 mm, 1 arc second is about 0.484814 um. | For fixture offsets or non-arc displacement, confirm whether chord, Abbe, or full kinematic modeling is required. | Derived from NIST radian definition | Model reviewed 2026-06-17 |
| Resolution is not accuracy | JCGM VIM defines resolution as the smallest measured-quantity change that causes a perceptible indication change; it can depend on noise or friction. | Fine controller counts or encoder interpolation cannot by itself prove installed rotary-axis accuracy. | JCGM 200:2012 (VIM) definition 4.14 | Published 2012; reviewed 2026-06-17 |
| Accuracy claim boundary | JCGM VIM defines measurement accuracy as closeness to a true value and notes that accuracy is not a quantity with a numerical value. | A supplier number labeled "accuracy" still needs test method, conditions, and uncertainty disclosure. | JCGM 200:2012 (VIM) definition 2.13 | Published 2012; reviewed 2026-06-17 |
| Uncertainty framing | NIST TN 1297 describes combined standard uncertainty as the method for combining individual standard uncertainties. | The page calculator is a screening tool. It is not a certified uncertainty statement. | NIST TN 1297 Section 5 | Reviewed 2026-06-17 |
| Expanded uncertainty | NIST TN 1297 defines expanded uncertainty U as combined standard uncertainty multiplied by coverage factor k. | Supplier comparisons need explicit k, confidence level, and test condition disclosure. | NIST TN 1297 Section 6 | Reviewed 2026-06-17 |
| High-resolution encoder counterexample | HEIDENHAIN lists RCN 2001/5001 classes with +/-2 arc-sec or +/-4 arc-sec system accuracy and 26-bit or 28-bit position values per revolution. | Even high position counts and low arc-second catalog classes still need installation-specific validation. | HEIDENHAIN angle encoders with integral bearing | Accessed 2026-06-17 |
Concept Boundaries for Decision Quality
VIM + NISTThe conversion answer is exact, but engineering decisions can fail when source-backed concepts are blended together. This table turns each concept boundary into a supplier or validation action.
| Concept | Source-backed boundary | Decision impact | Action |
|---|---|---|---|
| Second of plane angle | 1 second of plane angle = pi / 648000 rad. | Use this as the canonical conversion anchor for 1 arc second in radians. | Cite NIST SP 330 Section 4 in RFQ or documentation notes. |
| Radian projection | Plane angle in radian follows theta = s / r. | Use radians for circular arc length at a known radius; do not silently reuse it as chord or Abbe-offset displacement. | State radius, displacement model, and units beside every projected tolerance. |
| Resolution | Smallest measured-quantity change that causes a perceptible indication change. | Controller step size, encoder interpolation, and displayed decimals are not installed accuracy. | Request measured repeatability, reversal behavior, noise floor, and friction/load conditions. |
| Measurement accuracy | Closeness to true value; VIM notes accuracy is not itself a numerical quantity. | A single +/- number needs supporting test conditions before vendor ranking. | Ask for procedure, reference artifact, environment, load, speed, and uncertainty statement. |
| Expanded uncertainty | U = k x uc, with k chosen for the desired confidence level. | Two suppliers using different k values are not directly comparable. | Require coverage factor, confidence level, and combined standard uncertainty terms. |
Methodology
The method separates exact conversion from decision screening. That keeps the immediate tool result useful without overstating what a conversion can prove.
Core formulas
arcsec = value x unitFactor unitFactor: arcsec=1, degree=3600, radian=648000/pi, arcmin=60 degree = arcsec / 3600 radian = arcsec x pi / 648000 linear_shift_um = radian x radius_mm x 1000 tolerance_ratio = linear_shift_um / tolerance_um
The tolerance ratio is a screening output. Formal acceptance still requires measured uncertainty terms and documented test conditions.
Method flow
| Step | Formula | Output |
|---|---|---|
| Normalize input | arcsec = value x unitFactor | One common base unit for degree/radian conversion. |
| Convert to degree | degree = arcsec / 3600 | Exact decimal degree for CAD, spreadsheet, and RFQ notation. |
| Convert to radian | radian = arcsec x pi / 648000 | Required unit for arc-length projection and most engineering math. |
| Project to radius | shift_um = radian x radius_mm x 1000 | Practical estimate of linear shift at the working radius. |
| Compare boundary | ratio = shift_um / tolerance_um | Next action: proceed, validate first, or tighten specification. |
Comparison: Which Tool Belongs in Which Step?
Use this page for the conversion and early decision boundary. Use metrology reports for final proof.
| Option | Best for | Limitation | Next step |
|---|---|---|---|
| This canonical converter | Fast answer for arc second to degree, 1 arc second in radians, and radius projection. | Does not replace installed machine measurement or supplier acceptance testing. | Use before RFQ wording and early tolerance screening. |
| Generic scientific calculator | Raw arithmetic when the user already knows the formula. | Usually lacks unit context, alias answer, boundary notes, and rotary-axis next actions. | Use only for independent arithmetic spot checks. |
| CAD or motion-control software | Applying converted values inside a specific model or controller. | May hide unit assumptions or round values differently across systems. | Enter radians or degrees only after documenting conversion source. |
| Metrology acceptance report | Final proof of axis capability under test conditions. | Requires measured data, coverage assumptions, and documented procedure. | Use after conversion shows the target is relevant and tight. |
| Supplier catalog or encoder datasheet | Early screening of whether published classes are in the right order of magnitude. | Catalog accuracy, resolution, and repeatability can use different conditions and may omit coverage factors. | Use only as a shortlist input; require installed-axis evidence before final selection. |
Risks and Mitigations
The biggest risk is not the formula. It is using the formula without preserving unit context and measurement boundaries.
Unit risk
60x arc-min mismatch
Cost risk
Late redesign if accuracy is over-claimed
Scenario risk
Radius and temperature can change the decision
| Risk | Trigger | Impact | Mitigation |
|---|---|---|---|
| Unit mix-up | Arc-second, arc-minute, and degree values copied into the same RFQ without conversion. | A 60x error can enter the requirement before supplier review. | Normalize to arc-seconds and radians, then show the original unit in notes. |
| False capability claim | Exact conversion treated as proof that the machine can hold the angle. | Backlash, encoder, and thermal terms can exceed the converted value. | Pair conversion output with uncertainty budget and acceptance test evidence. |
| Radius mismatch | Linear shift calculated at 100 mm but applied to a larger fixture radius. | Linear error scales directly with radius. | Run the tool at worst-case radius and include the radius in the requirement. |
| Rounding loss | Small radian values shortened too aggressively in spreadsheets or CAD notes. | Downstream calculations can lose precision or become inconsistent. | Keep at least 9 significant digits for the 1 arc second radian anchor. |
| Model mismatch | Arc length is used as if it were chord displacement or probe error at an offset point. | The conversion can be numerically correct while the acceptance geometry is wrong. | Name the model: circular arc, chord, or installed fixture transform. |
| Uncertainty mismatch | Supplier claims are compared without coverage factor, confidence level, or test condition disclosure. | A cheaper axis can appear equivalent to a tighter measured system when the intervals are not comparable. | Ask for U, k, combined standard uncertainty terms, and repeatability conditions. |
Scenario Examples
Practical useEach example shows what to do with the converted value after the calculator returns a result.
Documentation answer
Premise: A reader asks "what is 1 arc second in radians?"
Process: Use the default tool preset and the direct-answer table on this page.
Outcome: Answer: pi/648000 rad, approximately 4.848136811e-6 rad.
RFQ tolerance translation
Premise: A buyer sees a 5 arc-sec rotary claim and needs CAD notation.
Process: Enter 5 arc seconds, convert to degree/radian, and keep the source identity in the RFQ.
Outcome: Supplier comparison starts from the same unit basis before accuracy evidence is requested.
Radius sensitivity check
Premise: An inspection station has a 300 mm working radius.
Process: Run 1 arc second at 300 mm and compare projected shift against the part tolerance.
Outcome: The team sees the linear effect triple versus the 100 mm baseline.
Arc-minute confusion prevention
Premise: A supplier sends 1 arc-min while the spec asks for 1 arc-sec.
Process: Use the conversion table to show 60 arc-sec = 1 arc-min.
Outcome: The team catches the 60x mismatch before purchase.
Acceptance-model selection
Premise: A probe checks a feature by straight-line endpoint displacement rather than swept arc length.
Process: Use the radian conversion as the angle input, then select chord or full transform before writing the inspection method.
Outcome: The conversion stays traceable, while the acceptance model matches the measured geometry.
Supplier evidence request
Premise: Two vendors publish similar arc-second claims but omit uncertainty details.
Process: Use the known-unknown checklist to request coverage factor, test conditions, repeatability setup, and installed-load data.
Outcome: Vendor ranking waits for comparable evidence instead of relying on label wording.
Known Unknowns
These are intentionally separated from conversion facts. Treat them as validation work, not as calculator output.
Installed-axis accuracy
Status: Not proven by conversion
Request measured backlash, repeatability, encoder, and thermal drift data.
Supplier confidence interval
Status: Often missing in catalog claims
Ask whether the value is standard uncertainty, expanded uncertainty, accuracy, repeatability, or resolution.
Application geometry
Status: Site-specific
Document working radius, load offset, and whether arc length or chord displacement is the right model.
Coverage factor and confidence level
Status: Pending confirmation unless supplier discloses k and interval basis
Request whether quoted uncertainty is standard uncertainty, expanded uncertainty, or an informal catalog limit.
Public proof of a specific installed machine
Status: No reliable public data for your installation
Treat catalog examples as class references only; validate with your fixture, load, temperature, and controller chain.
FAQ
What is 1 arc second to degree?
1 arc second to degree is 1/3600 degree, which is 0.000277777778 degree.
What is 1 arc second in radians?
1 arc second in radians is pi/648000 rad, approximately 4.848136811e-6 rad.
Is 1 arc second in radians the same intent as arc second to degree?
Yes. Both ask for the same angle conversion cluster, so this site keeps them on the canonical /learn/arc-second-to-degree page.
How many arc seconds are in one degree?
There are 3600 arc seconds in one degree. Divide arc seconds by 3600 to get degrees.
How many radians are in one arc second?
One arc second is pi/648000 rad, or about 0.000004848136811 rad.
How do I convert arc seconds to radians?
Multiply arc seconds by pi/648000. The tool does this automatically and also returns degrees and circular arc-length projection.
How do I convert radians to arc seconds?
Multiply radians by 648000/pi. Select radian as the input unit in the tool and it will return arc-seconds.
Why does the tool ask for radius?
The radius converts angular error into circular arc length: radians x radius mm x 1000 = micrometers.
Is the radius projection exact?
It is exact for circular arc length because radian is defined by arc length divided by radius. It is not automatically exact for chord displacement, Abbe-offset error, or a complete fixture transform.
Does conversion prove rotary table accuracy?
No. Conversion only changes units. Accuracy needs measured backlash, encoder, thermal, repeatability, and uncertainty evidence.
How many arc seconds are in one arc-minute?
There are 60 arc seconds in one arc-minute. This page includes the comparison because arc-min and arc-sec are commonly confused in RFQs.
What precision should I keep for 1 arc second in radians?
For engineering notes, keep at least 4.848136811e-6 rad or enough significant digits for your downstream tolerance stack.
Should supplier specs be compared in degrees or radians?
Use either after normalization, but radians are better for circular arc-length projection while arc-seconds are easier for precision rotary claims.
What is the difference between resolution and accuracy here?
Resolution is an indication floor, while measurement accuracy is closeness to true value and is not itself a bare numeric quantity in VIM terms. Do not treat encoder counts, display decimals, or interpolation as proof of installed-axis accuracy.
What should I ask a supplier after converting arc seconds?
Ask for measured repeatability, reversal behavior, load and temperature conditions, reference method, combined standard uncertainty, coverage factor k, and confidence level.
When should I mark the result as pending confirmation?
Mark it pending when the working radius, displacement model, installed backlash, thermal drift, measurement procedure, or supplier uncertainty basis is unknown. The conversion remains correct, but the capability conclusion is not proven.
What is the fastest way to catch a unit error?
Check whether the number is arc-sec, arc-min, or degree, then normalize everything to arc-sec and radians in one table.
Why not create a separate page only for 1 arc second in radians?
A separate page would duplicate the same conversion intent. Keeping it here strengthens the canonical answer and avoids internal competition.
Need supplier-ready angle conversion notes?
Share your arc-second target, working radius, and tolerance limit. We can turn the conversion output into a clearer RFQ or acceptance checklist.