How this calculator works
Light from the scene reaches the sensor through the lens, and the sensor width fixes how much of the scene fits across it. For a thin lens, the half angle is the arctangent of half the sensor dimension divided by the focal length, so the full angle is twice that. A longer focal length produces a narrower angle, which is why telephoto lenses magnify. The field of view on the subject is the chord across that cone at the subject distance, and for small angles it is very close to the straight-line width times distance divided by focal length. At three metres with a 50 mm lens and a 36 mm sensor, the recorded width is 36 x 3 / 50 = 2.16 metres.
Formula
horizontal angle = 2 x atan(sensor width / (2 x focal length)). The vertical and diagonal angles use the sensor height and diagonal. field of view = 2 x distance x tan(angle / 2), which is the same as distance x sensor dimension / focal length. For a chosen angle: focal length = sensor width / (2 x tan(angle / 2)).
Worked example
A 50 mm lens on a full-frame 36 by 24 mm sensor gives a horizontal angle of 2 x atan(36 / 100) = 39.598 degrees and a vertical angle of 2 x atan(24 / 100) = 26.992 degrees. The diagonal of the sensor is 43.267 mm, so the diagonal angle is 46.793 degrees. At 10 metres the horizontal field is 10 x 36 / 50 = 7.2 metres, the vertical field is 4.8 metres and the diagonal field is 8.653 metres. Asking for a 90 degree horizontal angle on the same sensor needs a focal length of 36 / (2 x tan 45 degrees) = 18 mm.
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Frequently asked questions
Why do the angles change when I change sensor size?
The angle depends on the sensor dimension as well as the focal length. A smaller sensor sees a narrower slice of the image circle, so the same lens behaves like a longer one; this is the crop factor effect.
Is the field of view a rectangle or a curve?
The true projection is slightly curved at wide angles, and this calculator uses the straight-line chord at the subject distance. For moderate angles the difference is tiny, but fisheye and very wide lenses need a projection-specific model.
Does focus breathing change the angle?
Yes. Many lenses change focal length as they focus closer, so the angle at a near subject can differ from the figure at infinity. The quoted focal length is normally the value at infinity focus.
How do I compare lenses across sensor sizes?
Use the crop factor from the sensor diagonal. A 25 mm lens on a two-times crop sensor frames like a 50 mm lens on full frame, and the field of view at the same distance is the same.