Degrees to Arc Seconds Converter for Optics and Lenses

Breaking a Lens Field of View Down to Arc Seconds

Optical data sheets describe the big picture in degrees — a lens covers 39.6° across the frame, a pan-tilt head sweeps 360°, a machine-vision camera sees a 25° cone. The small picture is always in arc seconds: how much of the scene one pixel subtends, how tightly a gimbal holds a target, how much residual tilt a mount leaves after alignment. Those are the same quantity at different magnifications, and the bridge between them is a factor of 3600.

Conversion factor: 1° = 60′ = 3600″, so degrees multiply by 3600. A 50 mm lens on a full-frame sensor covers 39.5978° horizontally, which is 142 552″; spread that across a 6000-pixel row and each pixel takes in 23.76″ of the scene.

Where the Fine Scale Matters

Angular size of a pixel

Dividing the field of view by the pixel count gives the angle one photosite covers. It sets how much detail the system can resolve regardless of how many megapixels are on the label.

Pointing and stabilisation specs

Gimbals, pan-tilt heads and image stabilisers are rated in arc seconds or fractions of a degree of residual error, because a tenth of a degree is already many pixels of smear on a long lens.

Machine-vision measurement

An inspection camera measuring a part at a fixed standoff converts angle to millimetres directly, so the per-pixel angle in seconds decides the smallest defect the station can call.

Mount and bench alignment

Wedge in a window, tilt in a mirror mount and parallelism between two optical faces are all specified in arc seconds, while the adjuster that corrects them is graduated in degrees.

Taking a Field of View Down to the Pixel

Start from the angle the data sheet gives, get it into seconds, and only then divide by whatever count the sensor or the encoder provides.

Enter the angle in degrees

Type the field of view, the tolerance or the sweep — 39.5978, 0.05, 25 — into the left field. Seconds appear as you type, with thousands spaced apart so a six-figure result stays readable.

Divide by the pixel or step count

Take the seconds figure and divide by the sensor's horizontal pixels, or by the steps in a rotation stage, to get the angle per element. That number is the honest resolution limit of the system before optics are considered.

Read the same angle in minutes when that reads better

Both dropdowns are searchable and carry every angle unit in the app, so a coarse figure that runs to hundreds of thousands of seconds can be shown in arc minutes instead without re-entering it.

Reverse it to size up a tolerance

Press swap (↔) when the specification arrives in seconds and the adjuster is marked in degrees — a 30″ wedge tolerance comes back as 0.008333°, which is what you dial against.

Angle per pixel is not the same as resolution. Two adjacent photosites cannot represent a feature one pixel wide — sampling theory asks for at least two pixels across the finest detail — and lens aberrations, diffraction and the anti-alias filter all soften what reaches the sensor. Treat the per-pixel angle as the best case the geometry allows, not as what the system will deliver.

Lens Fields of View Across a Sensor

Horizontal coverage for common focal lengths on a 36 mm-wide full-frame sensor, converted to seconds and then shared out over a 6000-pixel row. The pattern is the useful part: doubling the focal length roughly halves both the angle and the seconds each pixel receives.

Focal length	Horizontal field (°)	Same field (″)	Per pixel across 6000 px
14 mm ultra-wide	104.2500°	375 300″	62.55″
24 mm wide	73.7398°	265 463″	44.24″
35 mm reportage	54.4322°	195 956″	32.66″
50 mm standard	39.5978°	142 552″	23.76″
85 mm portrait	23.9132°	86 087″	14.35″
200 mm telephoto	10.2855°	37 028″	6.17″
600 mm super-telephoto	3.4367°	12 372″	2.06″

For comparison, an eye with 20/20 acuity separates detail about 60″ apart. Every row above is finer than that, which is why a print or a screen viewed at normal distance rarely shows what the sensor actually recorded.

What Helps at the Optical Bench

Step through a lens range in one sitting

The seconds figure follows every keystroke, so a set of focal lengths or standoff distances can be compared by overtyping the field instead of restarting the calculation.

Six-figure results stay legible

Thousands are separated by a space on screen, so a wide-angle field of a third of a million seconds does not arrive as an unbroken run of digits.

Tolerances read in either direction

Swap turns the pair round so a wedge or tilt figure quoted in seconds becomes the decimal degrees an adjuster or a CAD constraint is set in.

Numbers ready for the resolution sum

Copy returns the value with no unit and no spacing, so it goes straight into the division by pixel count without any cleaning up first.

Questions from the Optical Bench

Why are resolution figures in seconds when lens specs are in degrees?

Each unit is sized for the job it does. A field of view is tens of degrees, so degrees give a short readable number. A pixel on that same sensor covers tens of seconds and an alignment tolerance a handful, both of which would be strings of zeros in degrees. Using one unit for the whole optical path would force one end or the other into awkward notation, so the trade keeps both and converts where they meet.

How fine can a human eye resolve, in the same units?

Standard 20/20 acuity corresponds to separating two features one arc minute apart — 60″, or 0.016667°. Exceptional eyes reach perhaps half that under ideal contrast. It makes a handy sanity check on a display or a print: if a pixel subtends less than about 60″ at the intended viewing distance, adding more pixels stops buying visible detail. It is also why a 600 mm lens feels so different through a viewfinder, since it packs about thirty times the angular detail the unaided eye can pick out.

Does a longer lens give more seconds per pixel or fewer?

Fewer, and that is the point of a long lens. The pixel count stays fixed while the field of view shrinks, so each photosite covers less of the scene: 44.24″ on a 24 mm lens against 2.06″ on a 600 mm, a factor of more than twenty. The same relationship works against you for pointing — with each pixel worth two seconds, a vibration of a few tens of seconds smears the image across a dozen pixels, which is why long lenses demand stabilisation that wide lenses never need.

What does a 50 arc second pointing specification mean in millimetres?

It depends entirely on the range, since an angle only becomes a length once you fix a distance. Fifty seconds is 0.013889°, which puts a spot 24.2 mm to one side at 100 m and 0.24 m out at a kilometre. At a machine-vision standoff of 500 mm the same error is only 0.12 mm. Working the number back to a length at the distance you actually use is the quickest way to judge whether a pointing spec is generous or tight for the application.

How many decimal places of a degree does one second need?

Four at the very least, and five if the value will be used in further arithmetic. Three decimal places step in units of 3.6″, which is coarser than most alignment tolerances; four places step in 0.36″, comfortably below one second; five reach 0.036″. This matters when a CAD constraint, a controller field or an exported data file quietly rounds an angle — a mount specified to 5″ can be destroyed by a field that only stores three decimals of a degree.

Degrees to Arc Seconds