The binoculars range calculator above turns two numbers into a distance. You supply the real size of something you can see — a person, a vehicle, a window, a mast — and the angle it covers in the reticle of your binoculars, and the tool returns how far away it must be. This is stadiametric ranging, and it is the oldest optical distance method still in daily use, because it needs no laser, no battery and no second observer.
Arb Digital builds free physics calculators that each own one job properly. This page handles the size-and-angle route to a distance and, just as importantly, tells you how much that distance is worth. The uncertainty figure in the grid is the part most calculators leave out, and it is the part that decides whether the answer is usable.
What This Binoculars Range Calculator Does
Every angular measurement is a ratio. When an object of height H is far enough away that the angle it covers is small, that angle in radians is very nearly H divided by the distance. Rearranging gives the distance directly. Because a milliradian is one thousandth of a radian, working in mils turns the arithmetic into something you can do in your head: distance equals size in metres times a thousand, divided by the mil reading.
The calculator accepts four angular units because reticles disagree. True milliradians are what most modern ranging reticles use. NATO artillery mils divide the circle into 6,400 equal parts rather than the 6,283.19 that a true milliradian implies, a legacy of gunnery tables that wanted a round number, and they are about 2 per cent smaller. Minutes of arc are the American sporting convention, and plain degrees appear on some marine and surveying instruments.
The grid converts the range to yards, reports what angle your reading corresponds to in MOA so you can cross-check against a differently marked optic, gives the physical width that one mil covers at that range, and gives the range uncertainty implied by the reading tolerance you enter. The angular resolution calculator deals with the separate question of the smallest angle an aperture can distinguish at all.
How to Use It
- Pick a target whose size you actually know. Not one you can estimate. A known vehicle, a standard door, a survey pole. The output inherits your error in this number exactly, one for one.
- Measure the angle across the dimension you entered. If you entered a height, read the height. Mixing a width measurement with a height figure is the most common way to get an answer that is wrong by a factor of two.
- Select the correct angular unit. Mils and MOA differ by a factor of about 3.44, so choosing the wrong one produces a range that is wrong by that factor, not by a few per cent.
- Set the reticle plane and, if needed, the calibration power. A second-focal-plane reticle used away from its calibration magnification is measuring in units that are not the ones printed on it.
- Read the uncertainty, not just the range. If the tool says 400 metres plus or minus 9, you have a usable number. If it says 2,000 plus or minus 220, you have a rough bracket.
The Formula: How Stadiametric Range Is Calculated
The exact relation is D = H ÷ tan(θ), where D is the distance, H is the known size and θ is the angle it subtends. For the angles involved in practical ranging — a person at 400 metres covers about a quarter of a degree — the tangent is indistinguishable from the angle itself in radians, which is why the working form is D = H ÷ θrad. This calculator uses the tangent throughout, so it stays correct for close targets that cover a large angle, where the small-angle shortcut starts to drift.
The radian is defined as arc length divided by radius, which is precisely why this works: the ratio of a target's size to its distance is its angular size in radians. OpenStax University Physics Volume 1, section 10.1 on rotational variables, states the definition as θ = s ÷ r and is the cleanest derivation of the ranging identity you will find.
The unit conversions the tool applies are: one true mil is 0.001 radian exactly; one NATO mil is 2π ÷ 6400 = 0.00098175 radian; one minute of arc is π ÷ 10800 = 0.000290888 radian; one degree is π ÷ 180. The radian's status as the coherent SI unit of plane angle is set out in the NIST Guide for the Use of the International System of Units, Special Publication 811.
Work the defaults. A 1.8 metre target reads 4.5 true mils. That is 0.0045 radian, so D = 1.8 ÷ tan(0.0045) = 1.8 ÷ 0.0045000 = 400.0 metres, which is 437.4 yards. The same angle expressed in minutes of arc is 0.0045 ÷ 0.000290888 = 15.47 MOA. One mil at 400 metres covers 0.400 metres. A reading tolerance of ±0.1 mil puts the range between 4.4 and 4.6 mils, giving 391.3 to 409.1 metres — a spread of about ±9 metres, or 2.2 per cent.
Why the Error Grows Faster Than the Distance
This is the single most useful thing to understand about stadiametric ranging, and almost no calculator says it. The percentage error in your range equals the percentage error in your angle reading. That sounds benign until you notice that the angle shrinks as the target gets further away, while your ability to read the reticle does not improve.
Suppose you can reliably resolve a tenth of a mil. At 200 metres a 1.8 metre target covers 9 mils, so a tenth of a mil is about 1.1 per cent, and your range is good to roughly ±2 metres. At 800 metres the same target covers 2.25 mils, the same tenth of a mil is now 4.4 per cent, and the range is good to only ±36 metres. At 1,600 metres it covers 1.13 mils and the uncertainty is over 140 metres. The method has not got worse; the geometry has.
Two practical consequences follow. First, always range on the largest known dimension available, because a bigger angle carries proportionally less error. Ranging on a vehicle's length rather than its wheel diameter can cut your uncertainty by a factor of five. Second, average several readings if the target permits it, because reading error is largely random and averaging suppresses it, while a wrong assumed target size is a systematic error that averaging will never touch.
The Focal Plane Problem That Ruins Readings
Reticles live in one of two places inside an optic, and the difference decides whether your subtension values mean anything at a given magnification. In a first focal plane design the reticle sits ahead of the zoom group, so it magnifies along with the image. The target and the reticle grow together, the ratio between them never changes, and a mil is a mil at every power. Fixed-magnification binoculars behave the same way for the same reason: there is nothing to change.
In a second focal plane design the reticle sits behind the zoom group. It stays the same apparent size while the image grows around it, so a target covers more reticle divisions at high power than at low power even though nothing about the target has changed. The stated subtension is therefore correct at exactly one magnification, usually the highest, and the manufacturer prints that number somewhere in the manual.
The correction is a straight ratio: the effective angle equals the reading multiplied by the calibration magnification divided by the magnification actually in use. Read 4.5 mils at 5× on a reticle calibrated at 10× and the true subtension is 9 mils, halving your range. Select the second-focal-plane option above and the tool applies this automatically. This is the single most common source of a range that is wrong by a factor of two, and it is entirely avoidable.
Where Binocular Ranging Actually Gets Used
Most binoculars have no reticle at all, and this method needs one. Where you do find them is marine and military-pattern instruments, surveying and forestry glasses, and observation binoculars mounted on a tripod, because a handheld reading beyond about 5× is limited by your own tremor rather than by the optic. Mounting the instrument is usually worth more accuracy than upgrading it.
Marine use is the most forgiving case, because the sizes involved are genuinely standard. A container is 2.44 metres wide, a navigation buoy's published height is on the chart, and a lighthouse's focal-plane height is in the light list. Land use is harder because almost nothing is standard, and the temptation to assume a size is exactly where the method goes wrong. Forestry and survey work sidesteps this by carrying a graduated staff of known length and ranging on that, which turns an estimate into a measurement.
The technique also transfers to a camera. Once you know the sensor size, focal length and the pixel height of an object, you have an angle, and the same arithmetic applies. That is the same geometry the ground sample distance calculator uses to convert a camera's angular pixel size into metres on the ground, and the hyperfocal distance calculator works in the same optical world from the depth-of-field side.
Where This Sits Next to the Other Distance Tools
This page is the only tool on the site that gets a distance from an angle and a known size. The great circle distance calculator gets one from two sets of coordinates, the speed distance time calculator gets one from motion, and the 3D distance calculator gets one from Cartesian points. None of them touch optics.
On the optical side, the telescope magnification calculator works out the power an instrument delivers from its focal lengths, which is the magnification figure this page asks you for. The angular resolution calculator tells you the smallest angular detail an aperture can separate, which sets a hard floor on how finely any reticle can usefully be read. Use those two to characterise the instrument, and this page to use it.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Guessing the target size — the range is directly proportional to it, so a 10 per cent error in assumed height is a 10 per cent error in distance with no warning of any kind.
- Confusing mils with MOA — they differ by a factor of about 3.44. Choosing the wrong unit does not shift the answer slightly, it multiplies or divides it.
- Ignoring the focal plane — a second-focal-plane reticle read at half its calibration power reports half the true angle and doubles your range.
- Measuring height against a width figure — or ranging on a partly obscured target, where you are reading a smaller angle than the object really covers.
- Trusting a long range with a small reading — below about one mil the percentage error is large enough that the answer is a bracket, not a distance.
Related Free Tools From Arb Digital
Characterise the optic first with the telescope magnification calculator and the angular resolution calculator, which between them tell you what power you are working at and how fine a reticle reading can meaningfully be. For distances that come from somewhere other than an angle, use the great circle distance calculator, the 3D distance calculator or the speed distance time calculator. On the imaging side the ground sample distance calculator and the hyperfocal distance calculator apply the same optical geometry to cameras. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
Range in metres equals the target size in metres multiplied by 1,000 and divided by the reading in mils. A 1.8 metre target that covers 4.5 mils is at 1.8 times 1,000 divided by 4.5, which is 400 metres. The factor of 1,000 is there because a milliradian is one thousandth of a radian.
The percentage error in the range equals the percentage error in the angle reading, plus the percentage error in your assumed target size. Close in, where a target covers many mils, a careful reading gives a few per cent. Beyond about a kilometre with a person-sized target, the reading is worth only a rough bracket.
A true mil is one thousandth of a radian, so a circle contains 6,283.19 of them. The NATO artillery mil divides the circle into exactly 6,400 parts instead, which makes each one about 2 per cent smaller. Using the wrong one shifts a range by roughly 2 per cent.
About 3.4377. One mil is 0.001 radian and one minute of arc is 0.000290888 radian, and the ratio of those two numbers is where the figure comes from. This calculator converts between the two automatically, so you can read in whichever unit your reticle uses.
Only in a second focal plane optic. There the reticle stays the same apparent size while the image grows, so the subtension is correct at one stated magnification only. In a first focal plane or fixed-power instrument the reticle and image scale together and the reading is the same at every power.
Not with this method, which needs a measured angle. You can substitute a graduated staff held at the target, a camera image whose field of view you know, or a map fix, but a plain pair of binoculars gives you no angular scale to read against.
Because the small-angle approximation drifts for targets that cover a large angle. Below about 100 mils the two agree to better than a third of a per cent, but at close range the tangent form stays exact, and there is no cost to using it.
This tool is provided for educational and study use. It models ideal small-angle geometry and takes your stated target size and reticle calibration at face value, so treat its output as a physics estimate rather than a surveyed measurement. Navigation, surveying and any safety-critical distance should be confirmed with an instrument calibrated for the purpose.
