Calculating Object Diameter and Angle of View from Distance  Calculating Minimum Object Size and Magnification for Full Format  Calculating Minimum Object Size and Magnification for APSC Format
On this page, I demonstrate how you can calculate object diameter, minimum object size, and magnification of lenses for full and APSC format cameras/sensors. This page supplements pages where I determine this data for my Mmount lenses when used at the Ricoh GXR and the Leica M (Typ 240).
Sharpness and resolving crtiteria for lenses and sensors are often given in angular measurements. For example, the human eye is said to have a resolving power of one minute of an arc. This criterion was relaxed to 2 minutes of an arc for determining the circle of confusion (CoC), which is used in calculations of the hyperfocal distance (HFD) and the depth of field (DoF). But knowing an angle does not tell us the diameter of an object of that angular size at a certain distance (or simpler, how wide it is). In the following, I show how you can calculate the diameter at a certain distance.
The diagram below shows that the object diameter g is related to the angle of view a through the TAN function and scaled by the distance r:
Legend: r = (shortest) distance, g = object diameter, a = angle of view (alpha in diagram)
For the calculations, you need a calculator (or an Excel sheet). Here is an example table showing the diameters at different distances for the just mentioned 1 and 2 minutes of an arc, for 30' (the approximate angular size of the moon and the sun) and or 32' (the average angular size of the moon and the sun):
Distance 

Angle of View 
m 
km 

0.25 
0.5 
1 
2 
5 
10 
20 
50 
100 
1 
10 
100 
1000 
384,000 
149,600,000 

1'  0.073 
0.145 
0.291 
0.582 
1.454 
2.9 
5.8 
0.0145 
0.0291 
0.2909 
2.909 
29.09 
290.89 
111.70 
43,516.88 
2'  0.145 
0.291 
0.582 
1.164 
2.909 
5.8 
11.6 
0.0291 
0.0582 
0.5818 
5.818 
58.18 
581.78 
223.40 
87,033.75 
30'  2.182 
4.363 
8.727 
17.453 
43.634 
87.3 
174.5 
0.4363 
0.8727 
8.7267 
87.267 
872.67 
8726.70 
3,351.05 
1,305,514.57 
32'  2.327 
4.654 
9.308 
18.617 
46.542 
93.1 
186.2 
0.4654 
0.9308 
9.3085 
93.085 
930.85 
9308.49 
3,574.46 
1,392,550.09 
mm 
m 
km 

Diameter 
Legend:
Bold+italic: Viewing angle that corresponds to the CoC
Moon: distance = 384,000 km;diameter = 3,476 km; angular size ranges
from 29.2' to 33.3' nearly 32' on average
Sun: distance = 149,600,000 km; diameter = 1,392,684 km; angular size ranges
from 31.5' to 32.5', 32' on average
As you can see, the calculated values for diameters of the moon and sun come
close to the real values.
A viewing angle of 2' corresponds to the "acceptable sharpness" criterion for the CoC, which is based on the resolving power of the human eye (1/8 mm at 25 cm viewing distance), but somewhat relaxed. 1' would correspond more closely to the actual resolving power of the human eye.
Note that the resolving power of astronomical telescopes is also given as an angle. I will use this information to calculate, which details (measured in km) my telescopes can resolve under optimal conditions on the moon and on the sun. Perhaps, a moon map can help me to find out what this actually means...
In some cases, it might also be useful to know the viewing angle a for an object with diameter g at a distance r. The viewing angle a is given by the following formula:
Legend: r = (shortest) distance, g = object diameter, a = angle of view (alpha in diagram above)
For the calculations, you need a calculator (or an Excel sheet). Here is an example table showing the viewing angles in degrees (decimal...) for different distances and for different diameters in the millimeter range (which may correspond to aperture diameters):
Distance 

Diameter (mm) 
m 
km 

0.25 
0.5 
1 
2 
5 
10 
20 
50 
100 
1 
10 
100 
1000 

0.125  0.029 
0.014 
0.007 
0.004 
0.001 
0.0007 
0.0004 
0.0001 
0.0001 
0.000007 
0.000001 
0.00000007 
0.00000001 
0.145  0.033 
0.017 
0.008 
0.004 
0.002 
0.0008 
0.0004 
0.0002 
0.0001 
0.000008 
0.000001 
0.00000008 
0.00000001 
0.25  0.057 
0.029 
0.014 
0.007 
0.003 
0.0014 
0.0007 
0.0003 
0.0001 
0.000014 
0.000001 
0.00000014 
0.00000001 
0.5  0.115 
0.057 
0.029 
0.014 
0.006 
0.0029 
0.0014 
0.0006 
0.0003 
0.000029 
0.000003 
0.00000029 
0.00000003 
1  0.229 
0.115 
0.057 
0.029 
0.011 
0.0057 
0.0029 
0.0011 
0.0006 
0.000057 
0.000006 
0.00000057 
0.00000006 
1.5  0.344 
0.172 
0.086 
0.043 
0.017 
0.0086 
0.0043 
0.0017 
0.0009 
0.000086 
0.000009 
0.00000086 
0.00000009 
2  0.458 
0.229 
0.115 
0.057 
0.023 
0.0115 
0.0057 
0.0023 
0.0011 
0.000115 
0.000011 
0.00000115 
0.00000011 
3  0.688 
0.344 
0.172 
0.086 
0.034 
0.0172 
0.0086 
0.0034 
0.0017 
0.000172 
0.000017 
0.00000172 
0.00000017 
4  0.917 
0.458 
0.229 
0.115 
0.046 
0.0229 
0.0115 
0.0046 
0.0023 
0.000229 
0.000023 
0.00000229 
0.00000023 
5  1.146 
0.573 
0.286 
0.143 
0.057 
0.0286 
0.0143 
0.0057 
0.0029 
0.000286 
0.000029 
0.00000286 
0.00000029 
10  2.292 
1.146 
0.573 
0.286 
0.115 
0.0573 
0.0286 
0.0115 
0.0057 
0.000573 
0.000057 
0.00000573 
0.00000057 
20  4.581 
2.292 
1.146 
0.573 
0.229 
0.1146 
0.0573 
0.0229 
0.0115 
0.001146 
0.000115 
0.00001146 
0.00000115 
25  5.725 
2.864 
1.432 
0.716 
0.286 
0.1432 
0.0716 
0.0286 
0.0143 
0.001432 
0.000143 
0.00001432 
0.00000143 
50  11.421 
5.725 
2.864 
1.432 
0.573 
0.2865 
0.1432 
0.0573 
0.0286 
0.002865 
0.000286 
0.00002865 
0.00000286 
100  22.620 
11.421 
5.725 
2.864 
1.146 
0.5730 
0.2865 
0.1146 
0.0573 
0.005730 
0.000573 
0.00005730 
0.00000573 
Viewing Angle (Degrees, decimal) 
Legend:
Bold+italic: about 2' corresponding to the circle of confusion
Bold: about 1' corresponding to the limit of human resolviong power
red: about 1"
Notes: 1' = 0.016666, 2' = 0.033333 degrees in decimal notation
The table shows, among others, at about which distance the human eye can no longer resolve details of a certain diameter (this is reflected in the CoC, which is the basis for DoF scales on lenses and the hyperfocal distance). This information can also be used to estimate at what distance a disk of confusion, which is used in Merklinger's method, can no longer be resolved, albeit the theory, based on geometrical optics, would not take this into consideration.
The following calculations are useful for lenses. For each lens, an angle of view (for an 36 x 24 mm image format) and a minimum (or shortest) distance is given in the technical data. This allows you to calculate the minimum object size and the magnification for a lens using some geometric relationships.
The diagram below shows that the object size g is related to the angle of view a through the TAN function and scaled by the distance r:
Legend: r = (shortest) distance, g = diagonal object width, a = angle of view (alpha in diagram)
Since g is the diagonal object size, we need to calculate the width and height of the object rectangle. This is easy for the 3:2 image image format (w=3*a, h=2*a), which allow us to use the theorem of Pythagoras:
Replacing a by g, we arrive at the following fomulae for object width w and object height h:
Inserting the formula for g, finally leads to formulae in which object width w and object height h depend only on distance r and angle a::
Magnification is given by dividing the object width (36 mm for 35 mm film) through the image width w (analogously for the height h):
Take the inverse of M to arrive at the 1:X format:
Angle of view a = 90 degrees, a/2 = 45 degrees, tan(45 degrees) = 1
Shortest Distance r = 500 (mm), sqr(15) = 3.873
w = 6/3.873*500 = 774.4 mm; h = 4/3.873*500 = 516 mm
M = 0.046 => M = 1: 21.5
(No responsibility is taken for the correctness of this information)
In the case of an APSC (or better, DX format) sensor, you can either:
and then use the formulae for full format described above.
The crop factor for APSC (DX format) is typically given as 1.5. Since the sensor format is not exactly 24 x 16 mm (here, it use 23.6 x 15.7 mm, but I found several minor variants of this...), the second method would be a little more exact (1.525 for the width). The formula for the second approach uses the diagonal of the sensor and is as follows:
For the APSC sensor size given above, the diagonal is about 28.35 mm.
For the APSC sensor, I calculated the values from the same formulae, but used the adapted angle of view for the APSC format.