Field of View Calculator

Ask what a 50 mm lens sees and the only honest reply is another question. On what?

The number engraved on a lens describes the glass and nothing else. Focal length is an optical distance, not the length of the barrel: when a lens is set to infinity, it is the separation between the rear principal plane of the lens and the sensor. How much of the world ends up in the picture depends just as much on the rectangle of silicon or film that sits behind the lens. Change the rectangle and the same lens becomes a different window.

The angle is a ratio

Picture a pinhole in the front of a box and a sensor on the back wall. Light from the left edge of the scene crosses through the hole and lands on the right edge of the sensor. Draw that ray and the one from the opposite edge, and you have a triangle whose base is the sensor and whose height is the distance from hole to sensor. For a lens focused far away, that distance is the focal length. Half the sensor divided by the focal length is the tangent of half the angle, so the whole angle is twice the arctangent of the sensor dimension over twice the focal length.

Put numbers in. A 50 mm lens in front of a frame 36 mm wide and 24 mm high sees 39.6 degrees from side to side and 27.0 degrees from top to bottom. Measured corner to corner, across a diagonal of about 43.3 mm, the same lens sees 46.8 degrees. Three different answers, all correct, from one lens on one sensor.

The same triangle, continued out into the world, gives the size of the scene. At any distance the frame covers that distance times the sensor dimension, divided by the focal length. So at 10 m the 50 mm lens takes in a rectangle 7.2 by 4.8 m. A person 1.7 m tall, standing in it, fills a little over a third of its height. This is the calculation that answers the practical question, because a wall, a stage or a group of people is measured in meters, not degrees.

Which of the three angles do formats get compared by? By convention, the largest. The trade guideline for digital camera catalogs converts a focal length to its 35 mm camera equivalent through the ratio of the two image diagonals, and the 46.8 degrees that reference tables print first for a 50 mm lens on the 36 by 24 mm format is the diagonal figure and not the width. The diagonal is a fair way to compare lenses, but nothing in a photograph runs corner to corner. If you need to know whether a room fits, use the horizontal number.

Notice too that the relation is not a straight line. Halving the focal length does not double the angle. On the same frame a 24 mm lens sees 73.7 degrees across and a 12 mm lens 112.6, not 147. Only at the long end does the rule of thumb hold: 200 mm gives 10.3 degrees and 400 mm gives 5.15.

Crop factor, and what “equivalent” means

Sensors come in many sizes, and the names are looser than they sound. The frame called full frame is nominally 36 by 24 mm, the size of a 35 mm film frame, yet a reference table of sensor formats lists it as a range, 35.8 to 36 mm wide and 23.9 to 24 mm high, and two cameras are recorded at 35.9 by 23.9 and 35.9 by 24 mm.

The format called APS-C is not one size at all. Sensors sold under that name run from 20.7 by 13.8 to 28.7 by 19.1 mm, and are typically 22.3 by 14.9 mm from one maker and 23.5 by 15.6 mm from the others. Individual cameras are recorded at 23.5 by 15.7 and at 23.3 by 15.5 mm, and one maker describes its own version of the format as approximately 24 by 16.

A Four Thirds sensor is 17.3 by 13.0 mm, a one inch type sensor 13.2 by 8.8 mm, and a common medium format sensor 43.8 by 32.9 mm. Treat every preset as nominal.

The crop factor turns those sizes into one number: the diagonal of the 36 by 24 mm frame divided by the diagonal of the sensor in question. A 23.5 by 15.7 mm sensor has a diagonal of 28.26 mm, so its crop factor is 1.53. Makers of cameras in this class round the figure to 1.5, and with a multiplier of 1.5 a 50 mm lens is said to act like a 75 mm one. The Four Thirds sensor works out at 2.0, the one inch type at 2.73, and the medium format sensor at 0.79, which is a crop factor below one because the frame is larger than the reference.

Multiply a focal length by the crop factor and you get the focal length that would show the same diagonal angle on a 36 by 24 mm frame. That is all “equivalent” means here. A 33 mm lens on the smaller sensor frames a scene very nearly as a 50 mm lens does on the larger one, from the same spot. It is an equivalence of angle of view and nothing more. The arithmetic makes no statement about depth of field, exposure or noise, and it should not be read as one.

Even the angle matches only along the diagonal when the two frames have different shapes. A 25 mm lens on a 17.3 by 13.0 mm sensor and a 50 mm lens on a 36 by 24 mm frame both see 46.8 degrees corner to corner. Across, they see 38.2 and 39.6 degrees; up and down, 29.1 and 27.0. Cropping inside one sensor does the same thing. Cut a 16:9 video frame from the full 36 mm width and the height falls to 20.25 mm, so the vertical angle of that 50 mm lens drops from 27.0 to 22.9 degrees while the horizontal angle stays put.

Where the simple formula stops

The formula assumes the lens is focused at infinity. Focus closer and the lens moves away from the sensor, which makes the triangle taller and the angle narrower. For a simple lens the distance from lens to sensor becomes the focal length times one plus the magnification, where magnification is the size of the image divided by the size of the subject. That stretched distance replaces the focal length in the formula.

At ordinary distances the effect is small. A 50 mm lens focused on something 2 m away has a magnification of about 0.026, and its horizontal angle slips from 39.6 to 38.7 degrees. Up close it is not small at all. A 100 mm lens sees 20.4 degrees across a 36 mm frame at infinity. Focused to life size, with the subject 0.2 m from the lens and a magnification of one, the same lens sees 10.3 degrees. It has become, for framing purposes, a 200 mm lens.

That correction is itself an idealization. It holds for a thin, symmetrical lens. In a real compound lens the point the picture is projected through is the center of the entrance pupil, the aperture as seen from the front, and when the entrance and exit pupils differ in apparent size the factor becomes one plus the magnification divided by their ratio. Makers seldom publish that ratio, so a close-focus angle worked out from the marked focal length is an estimate, and a good calculator should say so.

The other assumption is hidden in the word rectilinear. An ordinary lens is designed so that the distance of an image point from the center equals the focal length times the tangent of the angle off axis. That rule keeps straight lines straight. It also sets a hard limit, because the tangent grows without bound as the angle off axis nears 90 degrees. A rectilinear lens can approach a view of 180 degrees and never reach it.

Fisheye lenses escape by following a different rule. In the equidistant mapping the image distance is the focal length times the angle itself; in the equisolid angle mapping it is twice the focal length times the sine of half the angle. The consequences are large. On a 36 by 24 mm frame a rectilinear 15 mm lens sees 110.5 degrees along the diagonal. An ideal equidistant 15 mm fisheye sees 165, and an equisolid one about 185. Real fisheyes follow their intended mapping only approximately.

So “15 mm” is not an angle either. It needs a sensor and a projection before it means anything.

Choosing a lens from the numbers

Turn the scene formula around and it picks the lens for you. The focal length that fits a subject is the distance times the sensor dimension, divided by the subject’s size. Suppose a wall 4 m wide must fit in the frame and you can stand only 3 m back. On a 36 mm wide frame that calls for 3 times 36 over 4, or 27 mm. On a sensor 23.5 mm wide it is 17.6 mm. This is why wide lenses are the usual choice for interiors and architecture: rooms rarely let you back up.

For a landscape it is easier to think in angles. A ridge that spans 60 degrees of the horizon needs a lens whose half angle is 30 degrees, which on the 36 mm frame is 18 divided by the tangent of 30 degrees, about 31 mm. Go much wider and the ridge fits with room to spare but shrinks: at 24 mm the frame at 10 m is 15 m across, and a 1.7 m figure in it is barely a sixth of the height.

Portraits run the other way. To give a standing person a frame 2 m high from 4 m away, a 24 mm high sensor needs 4 times 24 over 2, or 48 mm. Step back to 8 m and the same framing needs 96 mm. The framing is identical; the picture is not, because distance sets the perspective and the lens only decides how much of it to keep. From far away a scene looks flattened, and from close up it looks foreshortened.

The same proportion tells you how much detail you will get. A subject takes up the same share of the pixels as it does of the scene. If the sensor is 6,000 pixels wide and the frame at the subject’s distance is 7.2 m across, a bird with a half meter wingspan covers about 417 pixels.

What can these numbers not do? They cannot be more exact than their inputs, and both inputs are rounded. Sensor sizes are nominal, and so are focal lengths. Even the reference diagonal is rounded: angle of view tables take it as 43.3 mm, while the catalog guideline writes 43.27. Nor can they say anything about distortion, about how a particular lens changes its view as it focuses, or about how sharp the corners will be.

They do answer the question that matters before you carry a lens anywhere. Will it fit?