Free · Kinematics calculators — How things move ⏱️

Minute-of-Arc Angle, Span and Distance

Convert between minutes of arc, degrees, radians and milliradians, or solve for a positive distance or circular arc span with editable inverse inputs.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.

Formula and method

One minute of arc is one sixtieth of a degree. At 100 m, the retained radius-times-angle model gives an arc span of approximately 2.908882 cm.

s = Rθ, with θ in radians. One minute of arc = π/10,800 rad; R = s/θ and θ = s/R.

Worked example

Enter these known values and leave the other values blank.

Radius or viewing distance
100 m
Angular span
1 arcmin
Arc span or small-angle linear estimate
2.909 cm

Assumptions and limitations

  • Enter two known positive quantities and leave the third blank. All three results remain editable with independent unit selectors and locks.
  • The retained formula is exact for the length of a circular arc at radius R when the central angle is in radians. Using it for a straight span at a viewing distance is a small-angle approximation, not exact plane or chord geometry.
  • One degree contains 60 minutes of arc; a full turn contains 21,600 minutes of arc. This angular minute is not a minute of elapsed time. Milliradians use one thousandth of a radian.
  • For a centered straight chord at radius R, the exact span is 2Rsin(θ/2). For a centered span across a perpendicular plane at distance R, the exact span is 2Rtan(θ/2). Neither of those straight-span formulas is substituted into this retained workflow.
  • For wide angles, use the geometry that matches the actual measurement. The formula Rθ remains an arc-length relation; it should not be presented as an accurate wide-angle straight-span estimate.
  • Distances and spans use exact international inch/foot/yard/mile definitions. Mixed feet/inches and meters/centimeters controls retain their component order.
  • This workflow preserves the source positive-only model. Zero angle, zero distance or zero span is rejected; those boundaries can also leave an inverse undetermined. Locked contradictory values are reported.

Common questions

Is one minute of arc the same as one degree?

No. One degree contains 60 minutes of arc. One angular minute is π/10,800 radians.

Does the same angle cover a larger span farther away?

Yes. For this radius-times-angle model, doubling the distance doubles the span at a fixed angle.

Can I find angular span from distance and a known span?

Yes. Enter the two positive lengths in their selected units and leave angular span blank. The model computes θ=s/R and converts to the selected angular unit.

Is this an exact straight-span formula for large angles?

No. It is exact for a circular arc. A straight chord or a span in a perpendicular plane requires the corresponding sine or tangent geometry; Rθ approximates those spans only at small angles.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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