The field of view formula: exact arctangent, small-angle error and crop factor

The geometry

Put the lens at the apex of a triangle. A thin lens focused at infinity forms the image one focal length behind it, and the ray from the edge of the sensor back through the lens center fixes the angle. Half the sensor side and the focal length make a right triangle, so the tangent of half the field angle is (side ÷ 2) ÷ f. Doubling gives the formula that Edmund Optics’ imaging guide prints as its Equation 1:

field angle = 2 × arctan( sensor side ÷ (2 × focal length) )

Edmund adds that real lenses measure the focal length from the rear principal plane, which is rarely at the mount. For a telescope you use the published effective focal length, with any reducer or Barlow factor applied.

How wrong the small-angle shortcut gets

Astronomy texts often write s = F × θ, which the Sheffield optics notes use “given that θ is usually small”. Applied to a whole sensor, it says field angle ≈ side ÷ f radians, or × 57.3 for degrees. Tele Vue’s true-field formula for eyepieces, field stop ÷ focal length × 57.3, is the same approximation.

The shortcut always overstates the field, and the error depends only on the angle itself (computed):

Exact field angle	Shortcut gives	Overstatement
1°	1.000°	0.003 %
5°	5.003°	0.06 %
10°	10.03°	0.25 %
20°	20.21°	1.0 %
45°	47.47°	5.5 %
60°	66.16°	10.3 %
90°	114.59°	27.3 %

The shortest telescope in our data, a 178 mm astrograph, spans 11.55° across a 36 mm full-frame sensor, where the shortcut is off by 0.34 % (computed). Wide camera lenses for Milky Way work sit in the bottom half of the table. The field of view calculator uses the exact form everywhere, so the same tool works for a 2032 mm SCT and a 14 mm lens.

Width, height and diagonal

Apply the formula to each side separately. For the diagonal, compute the sensor’s diagonal in millimeters first, then take the arctangent. Combining the width and height angles with Pythagoras gives the wrong answer once the angles are large. On a 36 × 24 mm sensor at 14 mm, the field is 104.25° × 81.20° and the true diagonal is 114.18°; the Pythagorean shortcut on the angles would claim 132.14° (computed). With an APS-C IMX571 sensor at 400 mm the two methods differ by 0.001°, which is why the mistake goes unnoticed at telescope focal lengths.

Imaging area is pixels × pitch

The side in the formula should be the part of the sensor that records the image: pixel count × pixel pitch. Published sensor sizes are often rounded. ZWO’s ASI294 page lists the ASI294MC Pro as 19.1 × 13.0 mm with 4144 × 2822 pixels of 4.63 µm, while the product of those figures is 19.19 × 13.07 mm (computed). At 400 mm that changes the field width by 0.75′, and at 135 mm by 2.2′ (computed). The calculators use pixels × pitch whenever a camera’s maker publishes both numbers.

What crop factor is

Nikon defines its FX format as 36 × 24 mm, about the size of 35 mm film, and DX as 24 × 16 mm, which “introduces” a 1.5× crop factor: a 24 mm lens on DX gives about the view of a 36 mm lens on FX. The diagonals are 43.27 mm and 28.84 mm, a ratio of 1.50 (computed). The CIPA Exif 2.3 standard carries the same idea into image files with the tag FocalLengthIn35mmFilm, defined as the equivalent focal length assuming a 35 mm film camera. Both describe framing relative to a film frame. Neither describes the lens.

Why the optics stay the same

The Nikon Z 8 shows this on one body. Its full sensor is 35.9 × 23.9 mm with 8256 × 5504 pixels, and its DX image area records 5392 × 3592 of those same pixels. The pitch, derived from the width and pixel count, is 4.35 µm in both modes. With a 20 mm lens (all computed):

Mode	Recorded area	Field	Image scale
FX	35.90 × 23.90 mm	83.8° × 61.7°	44.85″ per pixel
DX	23.45 × 15.62 mm	60.8° × 42.7°	44.85″ per pixel

The DX field is narrower because the camera reads a smaller patch of the same image. Every pixel still covers 44.85″, and a star still drifts across the sensor at the speed set by the real 20 mm focal length. That matters for untracked exposure limits, which depend on the real focal length and the pixel pitch. Note also that the DX area measured in pixels is 23.45 mm wide, so the FX/DX ratio on this body is 1.53, not the nominal 1.5.

Crop factor also depends on which dimension you compare. For 3:2 sensors, width and diagonal agree. For other shapes they do not: an IMX533 square sensor (11.31 × 11.31 mm) has a crop factor of 2.71 by diagonal and 3.18 by width (computed). The target framing calculator avoids the issue by working from the real width and height.

Rectilinear lenses only

The arctangent formula assumes a rectilinear projection, in which straight lines stay straight. Edmund notes that the distortion of some short focal length lenses can change the actual field. A fisheye lens bends straight lines by design, so neither the exact form nor the shortcut describes it; use the maker’s published angle of view instead. The DSLR and mirrorless sensor table lists sensor dimensions for the bodies in our data.