Lens field of view: why sensor size and focal length set it

Mount a 16 mm lens and the picture looks far more zoomed in than expected — and the lens listing shows four different field angles for that one product. Those numbers are not settings; each one belongs to a sensor format. This guide shows where the angles come from, why nothing on the camera changes them, and how to turn the arithmetic around to pick a focal length that frames your part the first time. It is written for anyone specifying a lens for a machine-vision or board camera.

Why does one lens have a different field angle for every sensor size?

A lens projects one fixed cone of light behind it. The sensor crops a rectangle out of that cone, so a larger format catches a wider angle: the same lens spans 43.6° × 33.4° on a 1" sensor but only 21.8° × 16.2° on a 1/2.3" sensor.

The angle is pure geometry between the focal length and the sensor edge you measure to:

Use the sensor width for the horizontal angle, the height for the vertical, the diagonal for the diagonal figure. The formula holds for rectilinear lenses viewed from distances much larger than the focal length. Standard formats share a 4:3 shape, which fixes their dimensions:

That is why one listing prints several angles: the maker has run this arithmetic per standard format so you can read your sensor's row instead of computing it. The format ladder itself is explained in the 1/1.8", 2/3" and 1" sensor-format guide, and what moves when the same lens changes sensors is worked through in same lens, different sensor: how HFOV changes.

Can you adjust the field of view of a fixed focal length lens?

No — the angle is fully fixed by the two numbers you already chose, focal length and sensor. Focus travel moves the barrel a fraction of a millimetre to sharpen the image and never widens the frame, so the per-format angles in a lens table are sensor options, not settings on one camera.

To change the view you change a term in the formula: a shorter focal length, a larger sensor inside the lens's image circle, or a varifocal / zoom lens whose focal length adjusts over a range — the machine vision lenses catalog keeps a varifocal / zoom filter with 57 models. Where zoom pays off and where it costs sharpness is compared in the zoom working-distance guide; for the fixed-lens route, the focal-length selection guide walks the whole decision.

What field of view does a 16 mm lens give?

On a 1/2.3" sensor, such a lens sees about 21.8° × 16.2° — a 193 × 142 mm patch at 500 mm working distance (500 × 6.17 / 16 = 193). The table is the same lens on every standard format, so read your sensor's row.

Width at a distance drops the arctangent: width ≈ working distance × sensor dimension / focal length, exact within a couple of percent whenever the distance is many focal lengths long.

A C-mount lens from the catalog, specified for 1.1" sensors at 20 MP with F2.8-22 and <0.3% distortion:

How do you pick focal length for a target field of view?

Turn the formula around: focal length ≈ working distance × sensor width / target width. To frame 300 mm of a 1/2"-sensor scene (6.4 mm wide) from 600 mm away: 600 × 6.4 / 300 = 12.8 mm, so a 12 mm lens — the 16 mm would frame only 240 mm.

Check both stock candidates before ordering:

• 12 mm gives 600 × 6.4 / 12 = 320 mm, 7% more than the target; slide the camera in to 560 mm and the view trims to 560 × 6.4 / 12 ≈ 300 mm.
• 16 mm gives 600 × 6.4 / 16 = 240 mm, 20% short — and no camera position adds 60 mm back.

The decision: a 12 mm lens at a 560–600 mm working distance. Close-range work needs the exact thin-lens form, width = sensor × (working distance − focal length) / focal length; at 600 mm that puts the 12 mm view at 6.4 × 588 / 12 = 314 mm, within 2% of the simple answer. The same chain from stand-off to pixel size is applied end to end in the blister-inspection working-distance guide.

Does a shorter focal length always mean a wider view?

Yes — width at a fixed distance scales with 1/f: halve the focal length, about double the view. On a 1/2.3" sensor, the 16 mm focal length spans 21.8° × 16.2° and 6 mm about 54° × 42°, while each pixel's footprint grows 16 / 6 ≈ 2.7×, so fine details fade first.

The trade is real. Very short focal lengths bend straight lines at the frame edges and stretch the object-space detail thin, so wide lenses suit finding and guiding parts more than measuring them. Data sheets confirm the arithmetic: an 8 mm FA lens for 2/3" sensors lists FOV 57.6°, and 2 × arctan(8.8 mm / 16 mm) = 57.6° — the formula and the maker agree.

The catalog's 8 mm option for 2/3" sensors, FA type, with the angle printed as FOV 57.6°:

Board cameras make the spread vivid, because the M12 board lenses run from 2.85 mm to 25 mm:

Values as printed in each product's data sheet; open a part for its drawing and full specifications.

On a 1/2" sensor the pinhole arithmetic gives about 97°, 30° and 14.6° horizontal for those three focal lengths — the same scene in three frames, with distortion rising as the angle opens. Format matching, IR cut and the rest of board-lens choice are covered in the M12 lens selection guide.

When does the sensor-size rule stop applying?

Twice in practice: fisheye lenses, whose non-rectilinear projection breaks the arctan formula, and close-range or telecentric set-ups, where the view is sized by magnification — FOV = sensor dimension / magnification — not by an angle at all.

A rectilinear design stops covering a small format honestly below roughly 2 mm of focal length, which is why the M12 line includes a 1.45 mm fisheye for a 1/2.3", 10 MP sensor: read its projection curve instead of computing an angle. The fisheye selection guide covers circle, back focus and field angle for those lenses.

At close range, size the view from magnification instead: FOV = sensor width / m, and object space pixel size follows from the same division. The macro selection guide picks lenses that way, and telecentric designs go further and fix the window entirely — a designed object size at a fixed working distance, not an angle you compute.

One limit sits before the formula too: the image circle must cover the sensor diagonal with margin, or the corners render black. The diagonals are in the first table; the coverage-margins guide argues for how much extra to keep.

Which checks settle the choice before you order?

Five minutes of arithmetic decides it: compute the angle from sensor width and focal length, convert it to width at your working distance, check the image circle against the sensor diagonal, and confirm the smallest feature still spans about 3 pixels.

1. Angle: 2 × arctan(sensor dimension / (2 × focal length)) — the example focal length on a 1/2" sensor gives 22.6° horizontally.
2. Width: working distance × sensor width / focal length — 200 mm at 500 mm with that same lens.
3. Pixel footprint: divide the view width by the sensor's horizontal pixel count; keep the smallest feature you must resolve at 3 pixels or wider.
4. Circle: sensor diagonal plus margin, using the diagonal column of the first table.
5. Reality check: photograph a ruler in the first captured frame and back out the true angle, as calculating camera angle of view from calibration shows.
6. Mechanics: confirm mount and back focus; on S-mount barrels, the M12 focusing procedure sets and locks focus without drift.

When the numbers land, the lens finder filters by sensor format, focal length and resolution, and the lens configurator matches a lens to a chosen camera step by step. If the view maths still fights you, send the applications team the sensor, the part size and the stand-off — those three numbers settle most lens questions in one reply.