# How to determine via a math formula when a lens is wide angle or macro

Lets say I have a device with 3 separate lenses. One will be the main reference lens, and then the two others may be a wide angle lens or a macro lens, or also even a replica of the main reference lens and not be different at all.

How can I calculate if any of the 2 extra lenses is a wide angle or a macro lens?

The way I am doing it now is very rough and probably totally wrong. I use the next formula to calculate the field of view for each lens taking into account that I have the sensor size width and the focal length for all of them:

field_of_view = 2 * arc_tangent(sensor_size_width / (2 * focal_length))


Then I compare the obtained field of view against the reference lens one, and if I find that such field of view is around 10% larger then I assume is wide angle, and if is 10% less smaller I assume is a macro lens.

Probably my way to do it is totally wrong, so any tips or corrections will be so much appreciated.

• I think you may be a little confused here. "Macro" is not directly related to the focal length and/or field of view of a lens, but instead is about the maximum magnification a lens can produce, which needs to include its minimal focus distance; did you mean "telephoto" rather than "macro"? Jul 9, 2021 at 11:10
• @PhilipKendall Thanks for your reply. To solve this problem I have a mobile device which has 3 lenses: 1 main lens with a 1.2 fov, a wide-angle lens with a 1.7 fov, and a macro lens with a 0.6 fov. I don't have other devices to compare against, so I don't know if these field of view are just circumstantial, or can be used to produce a generic formula which will allow me determine when a lens is wide angle or macro in any device, as all devices will have different lens specs with different field of views. Jul 9, 2021 at 11:21
• The relationship between focal length/field of view and minimum focusing distance, the latter of which is what defines a macro lens, is purely coincidental. Jul 9, 2021 at 12:09
• @MichaelC Thank you, I understand it. So, I assume there is no proper way to find out when a lens is macro. About the wide-angle, is there some generic rules I could use to determine when is a wangle angle lens? Jul 9, 2021 at 12:38
• Traditionally, in photography a normal lens is one which in which the focal length is approximately the same length as the diagonal of the film or sensor size. A wide angle lens is anything with a focal length approximately 0.67X or less of the diagonal measurement, and a narrow angle (or "telephoto") is a lens with a focal length at least 1.5X the diagonal. Jul 9, 2021 at 15:28

For valid reason, which I will explain later, the “normal” focal length for any size camera is a focal length that is approximately the same length as the diagonal measure of its format size. Additionally, the aspect ratio of most rectangular format cameras is 3÷2 = 1.5. In other words, the length is 1.5 times the height.

If you mount a lens, focal length equal to the diagonal measure, the angle of view delivered is 53°. This is the most often quoted angle of view, somewhat like the fact that TV sets are sold by their diagonal measure.

If the camera’s aspect ratio is 3:2, and the lens focal length equals the diagonal measure, the angle of view, camera held horizontally will be approximately 45°.

A lens is generally considered to be wide-angle if its focal length is about 70% of “normal” or shorter. A lens twice “normal” or longer falls into the realm of telephoto.

Why is a “normal” lens one that is about equal to the hypotenuse of the format rectangle?

All lenses project a circular image. Only the central portion of this image is suitable photographically. Additionally, this image is vignetted (a dimming of the edges). The central portion is called “the “circle of good definition”. For these reasons, the camera masks off this circular image, creating a rectangular format or frame size.

Now the rest of the story: We see with our eye/brain combination. We perceive what we call “the human perspective”. This is the 3D vision we are blessed with. We judge the relationships of objects by how we perceive them as to their size and distance. A “normal” lens is one that roughly replicates the “human perspective”.

If we were to view a vista through a glass windowpane, we can trace the outline of objects on the glass with wax pencil. We can also image this scenery using a camera. We then view the picture we have made. If we view this image from a distance equal to the focal length of the taking lens, we will perceive that this image replicates the “human perspective”.

Such a viewing distance is likely impossible given that today’s cameras are small devices fitted with short lenses -- plus the closest we can focus with our unaided eye is 6 inches / 150mm. Also, we tend to look at pictures from a distance about equal to their diagonal measure.

Suppose you take a picture with a full frame 35mm camera and view an 8x10 print made from that image. The 8x10 inch print is an enlargement, magnification 8X. To view a picture and perceive via the “human perspective”, the math is --- viewing distance is the focal length of taking lens multiplied by magnification. For this image, the viewing distance for a “normal” 50mm lens is 50 X 8 = 400mm = 16 inches, about normal reading distance. The APS-C requires 12X magnification, the normal lens is 30mm thus 30 X 12 = 360mm = 14 inches viewing distance. See how this all works out to the “human perspective”.

The macro lens is one that is optimized to work in close, we are talking live-size = 1:1 meaning magnification 1. No such formula, thus the lens specifications.

• I think using 4:5 image dimensions for 2:3 sensors confuses things a bit. And using FL x enlargement is more complicated than need be (and probably erroneous because stated FL's are almost always approximate). Jul 10, 2021 at 14:45
• Had I used an 8X12 display image, we are talking a diagonal of 14.4 inches = 366mm, the typical viewing distance. The 8X10 typical viewing distance is 360mm. In other words, nothing much changes. By the way, perspective linked to focal length and the math related is tried and true. Jul 10, 2021 at 14:57
• The diagonal (typical normal viewing distance) of an 8x10 is 12.8", you calculated it at 16" for 35mm and 14" for APS. 14" is closer, but it should have been a 29mm lens actual... whatever is on the lens barrel probably isn't actual. The discrepancy is because the print is a crop, and math is required when the image is cropped to less than 53˚. But the math you used would not work for an 8x10 if the image was cropped to less than full height; you would have to figure out how much of the original image area remained instead. When you start including cropping, all of the "normals" change. Jul 10, 2021 at 15:27
• @StevenKersting Many phone cameras, which is the context of the OP are 4:3, not 3:2. Jul 11, 2021 at 3:38
• @MichaelC, still not 5:4 (8x10) w/o cropping. Jul 11, 2021 at 13:08

Macro is defined by magnification = (projected image size on the sensor) / (actual object size).
Macro magnification is generally considered 1.0, but can be greater than 1.0 or slightly less than 1.0. Most decent quality macro lenses have flat field focus (handy for copy work) and very little distortion. Common focal lengths, with respect to 35mm sensors, for macro lenses range from 40mm to 200mm with 100mm being very common. A zoom lens with macro stamped on it isn't a macro lens, but does have close focusing capability.

As for classifying if the lens is telephoto, normal, wide, ...; a chart commonly found on the Internet can be used as a handy guide. The various classifications are delineated by angular field of view.

This particular image was lifted from this web site.

• You probably should "translate" everything in your answer that applies to FF sensors or 135 film so that it is correct for the much smaller sensors that most phones have, which is specifically what the OP is asking about. Jul 10, 2021 at 5:51
• @MichaelC OP does not say which camera (cell phone) they own. The OP understands the relationship between FOV, sensor size, and focal length. The takeaway from the chart is the lens classification versus the FOV.
– qrk
Jul 10, 2021 at 18:41

There are several misnomers being used that confuse things... i.e. a telephoto lens is a lens design that has little to do with it's FOV; the same is true for a macro lens. But your idea of calculating/comparing the recorded FOV is valid in terms of determining if the lens used is wide/normal/narrow.

Technically there are short focus lenses, and long focus lenses; short focus lenses are often referred to as wide angle, and long focus lenses are often of telephoto design.

A lens is short focus when its' focal length is less than the diagonal measure of the imaging area/sensor; it is long focus if the FL is greater than the diagonal. And then there is the normal lens, which is a lens who's focal length is the same as the image diagonal, and which is the standard/reference you would compare against.

In all cases the normal lens has an ~53˚ FOV; because, regardless of the image size/format (large format/cell phone, rectangular/square), it requires an ~53˚ circular FOV (image circle) to cover its' longest dimension/diagonal. And that will then appear correct/normal when viewed at a distance equal to the output image's diagonal measure, i.e. occupying ~53˚ of your FOV and undistorted (which correlates closely with the human's ~60˚ visual field of object recognition).

However, having a focal length exactly matching the imaging area diagonal is unlikely. And there is a lot of rounding of numbers in most things photography related. So there are a lot of generalizations made for "close enough." E.g. a normal lens is often considered to be any lens with a circular FOV near 53˚ (~ +/- 7˚).

This chart from an article at the Pentax forums groups lens focal lengths by format size, but more importantly it also gives the circular (diagonal) FOV's. You could use this chart (or something similar) to categorize your calculated image FOV's. Anything labeled wide is short focus, and anything labeled tele is long focus (it may/may not actually be of telephoto design); but all of the divisions/groupings are somewhat arbitrary. This chart has groupings in ~ 15˚-20˚ FOV based on common focal lengths... you may find/hear even more arbitrary assignments.

(focal length and focus distance are the same measurement in this context; it is the distance between the lens' primary principle point and the image plane when the image is in focus)