In the most recent update of this post:
Most universal types of binoculars
I wrote, among others, this fragment (FOV means Field of View and FOV-deg is the FOV given in degrees):
“Sometimes the FOV in binoculars is given in meters at 1000 meters (FOV-m), but you can convert it to FOV-deg without any problems:
FOV-deg = FOV-m * 57/1000
FOV-m = FOV-deg * 1000/57
I call it the “magic number 57”, which gives VERY good approximations. I will write about this “magic number 57” in another post.
Let's analyse some real examples of binoculars from the same producer:
Binoculars Nikon Aculon 7x35 (9.3 deg or 163m/1000m):
(…)
FOV-deg = 163 * 57/1000 = 9.291
FOV-m = 9.3 * 1000/57 = 163.158
Binoculars Nikon Aculon 8x42 (8.0 deg or 140m/1000m):
(…)
FOV-deg = 140 * 57/1000 = 7.980
FOV-m = 8.0 * 1000/57 = 140.351
Binoculars Nikon Aculon 10x42 (6.0 deg or 105m/1000m):
(…)
FOV-deg = 105 * 57/1000 = 5.985
FOV-m = 6.0 * 1000/57 = 105.263”
So, now I will describe how you can “find” this “magic number 57”.
The full angle is 360 degrees, right? Right.
Imagine that you are standing in the middle of a building with a perfectly round wall (a kind of silo) where the length of the wall is exactly 360 meters.
When your FOV is 1 degree then you will see exactly 1 meter, right? Right, but this 1 meter is NOT in a straight line, but it is a little curved. We have to find an approximation of a straight line.
The obvious question is this: “What is the distance from your eyes to the wall?”
360 meters = 2 * Pi * r
r = 360 meters / (2 * Pi)
The number Pi has to be rounded, so everything is rounded, but I round the number Pi at 4th decimal place, so all my calculations are VERY precise.
r = 360 meters / (2 * 3.1416)
r = 360 meters / 6.2832
r = 57.296 meters
On a side note: this value but with degrees instead of meters is equal to 1 radian, which is a a unit for measuring angles.
Now let's analyse the value of 2 * Pi (6.2832). This value means that in the full angle there are slightly more than 6 equal angles of 57.296 degrees (360 degrees / 57.296 degrees = 6.2832).
This means that as an approximation we can use a regular hexagon that consists of 6 regular triangles. Actually we can use 2 different hexagons: a hexagon in a circle or a hexagon on a circle. The more I think about it the more I believe that a hexagon on a circle should be used, but it's a more complicated case. Let's start with the more natural case.
1. Case #1 – hexagon in a circle.
A regular hexagon in a circle with a radius of 57 meters consists of 6 regular triangles with sides of 57 meters:
Obviously the perimeter of the hexagon is 342 meters (6 * 57 meters).
It's 0.95% of the length of the wall of the silo (342 meters / 360 meters = 0.95).
The error is only 5% in the full circle. It's a VERY good approximation! So, we can consider that 1 meter of the perimeter of the hexagon is a very good approximation of 1 degree.
To calculate the FOV in meters at 1000 meters we need a regular triangle with sides of 1000 meters. We have 6 triangles with sides of 57 meters, so we can use one of them and convert it. This conversion explains my formula for FOV-m given above.
In the regular triangle with sides of 57 meters 1 degree is represented by 1 meter, so in the regular triangle with sides of 1000 meters 1 degree is represented by 17.544 meters (1 * 1000/57).
FOV-m = FOV-deg * 1000/57
By reversing the formula you get this formula:
FOV-deg = FOV-m * 57/1000
Magic number 57.
HOWEVER, there are 2 problems with THIS case (a hexagon IN a circle). The most important thing is the fact that the shortest distance to the crucial line (marked as x on the picture below) is LESS than 57 meters (LESS than 1000 meters when converted):
The second problem is the fact that (in THIS case) as an approximation of the FOV-m we use one of the sides of the triangle, so the FOV-m by definition can't be over 1000 meters at 1000 meters, which, I think, is actually NOT correct!
2. Case #2 – hexagon on a circle.
A regular hexagon on a circle with a radius of 57 meters consists of 6 regular triangles with sides of MORE than 57 meters:
We know that r is equal to 57 meters, so we can calculate a:
a = 57 meters * 2 / 1.732 = 65.82 meters.
Obviously the perimeter of the hexagon is 394.92 meters (6 * 65.82 meters).
It's 110% of the length of the wall of the silo (394.92 meters / 360 meters = 1.097).
The error is 10% in the full circle, so it's somewhat significant, BUT smaller angles would cause smaller errors! Already the angle 30 degrees (half of 60 degrees) gives a MUCH better approximation!
You can clearly see that within the orange angle the curved wall of the silo is VERY close to the crucial orange bottom line.
Before we do calculation for a smaller angle, it's good to finish this case, so we know which FOV-m is precise for the angle of 60 degrees.
Let's remind the crucial picture:
We have a regular triangle with the sides of 65.82 meters, that represents 60 degrees, BUT the distance to the crucial line is 57 meters! We want to calculate the FOV-m at 1000 meters, so we have to convert 57 meters to 1000 meters (1000/57). And we get the same formulas!
FOV-m = FOV-deg * 1000/57
FOV-deg = FOV-m * 57/1000
Magic number 57.
My approximation in the small triangle (before conversion) was 1 meter for 1 degree (assuming the 10% error), so after conversion the FOV-m for the angle of 60 degrees is equal to:
FOV-m= 60 * 1000/57 = 1052.63 meters at 1000 meters
The precise FOV-m for the angle of 60 degrees is this:
FOV-m-precise = 65.82 * 1000/57 = 1154.74 meters at 1000 meters.
3. Case #3 – dodecadon on a circle.
A dodecadon has 12 sides, so twice as many as in a hexagon.
Just looking at the picture above is enough to realise that the approximation for the angle of 30 degrees will be MUCH better!
We know that r is equal to 57 meters, so we can calculate a:
a = 57 meters * 2 / (1.732 + 2) = 30.55
Obviously the perimeter of the dodecadon is 366.60 meters (12 * 30.55 meters).
It's 102% of the length of the wall of the silo (366.60 meters / 360 meters = 1.0183).
The error is ONLY 2% in the full circle, so it's a FANTASTIC approximation!!! So, we can consider that 1 meter of the perimeter of the dodecadon is a fantastic approximation of 1 degree!!!
We have a non-regular triangle with the bottom side of 30.55 meters, that represents 30 degrees, BUT the distance to the crucial line is 57 meters! We want to calculate the FOV-m at 1000 meters, so we have to convert 57 meters to 1000 meters (1000/57). And we get the same formulas!
FOV-m = FOV-deg * 1000/57
FOV-deg = FOV-m * 57/1000
Magic number 57.
My approximation in the small triangle (before conversion) was 1 meter for 1 degree (assuming the 2% error), so after conversion the FOV-m for the angle of 30 degrees is equal to:
FOV-m= 30 * 1000/57 = 526.32 meters at 1000 meters
The precise FOV-m for the angle of 30 degrees is this:
FOV-m-precise = 30.55 * 1000/57 = 533.96 meters at 1000 meters.
Obviously the smaller the angle the better the approximation.
On some Internet site I found a calculator that calculates FOV-m from FOV-deg that uses the tangent function, but this is a much WORSE approximation!
FOV of 60 degrees is converted to 1732.05 meters at 1000 meters !?!?!?!?!?
The error on that Internet site for this angle is 50% (1732.04 / 1154.74 = 1.500) !?!?!?!?!?
In my approximation the error for the angle of 60 degrees was “only” 10%.
With smaller angles the error on that Internet site gets smaller, but it's still much bigger than the error in my approximation, for example:
FOV of 30 degrees is converted to 577.35 meters at 1000 meters.
The error on that Internet site for this angle is still quite significant: 8% (577.35 / 533.96 = 1.081), while the error in my approximation is only 2%!
Please notice that human eyes have different FOV for different purposes, but everything above 60 degrees is usually considered peripheral vision.
Please notice that the magic number 57 is true not only in the metric system, but also in the yard system! For example 1 degree is given as 52.5 feet at 1000 yards:
1 degree = 52.5 feet at 3000 feet
3000 feet / 52.5 feet = 57.1429
HOWEVER the formulas have to be slightly different:
FOV-f = FOV-deg * 1000*3/57
FOV-deg = FOV-f * 57/1000/3
Let's analyse a real examples of binoculars: Binoculars Nikon Aculon 8x42 (8.0 deg or 420f/1000y):
FOV-f = 8.0 * 1000*3/57 = 421.053
FOV-deg = 420 * 57/1000/3 = 7.980
Magic number 57.
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