The data states the diagonal angle of view is 40° for this 40mm. This is true if the lens is mounted on a DX (compact digital). This 40° angle of view is computed based on the diagonal measure of this format. The DX format measures 16mm vertical (height) --- 24mm horizontal (length) – 28.8mm diagonal (corner to corner).
We can use ordinary lens formulas to compute the angle of view realized. Below is the formula you can plug into Excel. A quirk of Excel is it works in radians not degrees. At the end of the formula we plug in the conversion which is *180/PI().
Anyway the math is:
Diagonal angle of view =((ATAN(28.8/2/40)))*180/PI()*2
Vertical angle of view =((ATAN(16/2/40)))*180/PI()*2
Horizontal angle of view =((ATAN(24/2/40)))*180/PI()*2
Answer for 40mm lens 16mm vertical dimension angle of view = 22.6°
Answer for 40mm lens 24mm horizontal dimension angle of view = 33.4°
Answer for 40mm lens 28.8mm diagonal dimension angle of view = 39.7°
OK, the angle of view is computed based on the focal length and one of the three dimensions of the format. Now, keep in mind that the focal length is computed based on the lens imaging an object at infinity ∞. This will be objects about 3000 X the focal length away from the camera thus 40 X 3000 = 120,000mm = 4,724 inches = about 400 feet distant. In other words, the camera lens projects an image of the outside world on the surface of film or digital sensor. The distance lens to image is published as the focal length, and this value is only true if the lens is imaging a distant object.
As you image objects closer than infinity ∞, the distance lens to image increases. At life-size (magnification one also called unity or 1:1) the projection distance is 2X the focal length. In other words at unity, a 40mm lens will project from a distance of 80mm. For a unity setup, the focal length, now called the “back focus” is 80mm. Now we must compute using 80mm. For this setup:
Answer for 80mm lens 16mm vertical dimension angle of view = 11.4°
Answer for 80mm lens 24mm horizontal dimension angle of view = 17.1°
Answer for 80mm lens 28.8mm diagonal dimension angle of view = 20.4°
You can compute the size of the image based on the fact that the image triangle is similar to the object triangle. The length of the base of the object triangle is the distance lens to object. The length of the base of the image triangle is the distance lens to image. All angles are equal and all sides have the same ratio.
Another approach is to calculate the object distance that yields a specific magnification. Suppose, using an DX camera, format 16mm height by 24mm length with a 40mm lens mounted.
Our objective is to copy a document 5 inches by 8 inches and fill the frame to the maximum.
We calculate the magnification needed for both height and length, we must use the minimum of these two values otherwise we will get overspill.
The magnification required for the short distance is 16 ÷ 127 = 0.126 written as 0.126X
The magnification required for the long distance is 24 ÷ 203 = 0.1182 written as 0.1182X
We use the lower of the two = 0.1182X
Now the task is to setup a camera to object distance that yields a magnification of 0.1182X
If we use 0.1182 X then the image dimensions of the object will be:
127 X 0.1182 = 15mm
203 X 0.1182 = 24mm.
To achieve, using a 40mm lens, we now calculate object distance.
m = magnification
f = focal length
p = object distance
m =f ÷ (p –f )
0.1182 = 40 ÷ (p – 40)
p = 378.4mm
Object distance 378.4mm with a 40mm yields the required magnification. The object fills the camera frame.