Stadiametric rangefinding, or the stadia method is a technique of measuring distances with a telescopic instrument. The term stadia comes from a Greek unit of length Stadion (equal to 600 Greek feet, pous) which was the typical length of a sports stadium of the time. Stadiametric rangefinding is used for surveying and in the telescopic sights of firearms, artillery pieces, or tank guns, as well as some binoculars and other optics. It is still widely used in long-range military sniping, but in many professional applications it is being replaced with microwave, infrared, or laser rangefinding methods. Although much easier to use, electronic rangefinders can give away the shooter's position to a well-equipped adversary, and the need for accurate range estimation existed for much longer than electronic rangefinders small and rugged enough to be suitable for military use.
The stadia method is based upon the principle that in similar triangles homologous sides are proportional. This means that, for a right triangle with a given angle, the ratio of opposite side length to adjacent side length (see tangent) is constant. By using a reticle with marks of a known angular spacing, the principle of similar triangles can be used to find either the distance to objects of known size or the size of objects at a known distance. In either case, the known parameter is used, in conjunction with the angular measurement, to derive the length of the other side.
Since a radian is defined as the angle formed when the length of a circular arc equals the radius of the circle, a milliradian (sometimes called a mil, or mrad), is the angle formed when the length of a circular arc equals 1/1000 of the radius of the circle. An object 5 meters high, for example, will cover 1 mrad at 5000 meters, or 5 mrad at 1000 meters, or 25 mrad at 200 meters. Since the radian expresses a ratio, it is independent of the units used; an object 6 feet high covering 1 mrad will be 6000 feet distant.
In Practice, it can be seen that rough approximations may be made with a right triangle whose base (b) is equal to the distance of the 'rangefinder' from the eye; with the aperture (a) being the hole through which the target is sighted - the apex of this triangle being on the surface of the user's eye. For a standard distance from the eye (b) of 28" (71.12 cm); this being the common length of an archer's draw: 28" × 1 milliradian ≈ 0.028" (0.071 cm) 10 milliradians ≈ 0.280" (0.711 cm) 100 milliradians ≈ 2.80" (7.112 cm)
The approximate range of an object 12" (30.48 cm) in height covering roughly 100 milliradians (2.80" or 7.112 cm) is 10 feet (3.048 m) or: Range (r) = approximate height of object (h) × (1000 ÷ aperture in milliradians (a)) r = h(1000/a) → where r and h are identical units, and a is in milliradians. The above formula functions for any system of linear measure provided r and h are calculated with the same units.
Stadia readings used in surveying can be taken with modern instruments such as transits, theodolites, plane-table alidades and levels. When using the stadia measuring method, a stadia rod is held so that it appears between two stadia marks visible on the instrument's reticle. The stadia rod has measurements written on it that can be read through the telescope of the instrument, providing a known length for the distance calculations.
An instrument equipped for stadia work has two horizontal stadia marks spaced equidistant from the center crosshair of the reticle. The interval between stadia marks in most stadia instruments gives a stadia interval factor of 100. The distance between the instrument and a stadia rod can be determined simply by multiplying the measurement between the stadia hairs (known as the stadia interval) by 100.
The instrument must be level for this method to work directly. If the instrument line of sight is inclined, the horizontal and vertical distance components must be determined. Some instruments have additional graduations on a vertical circle to assist with these inclined measurements. These graduated circles, known as stadia circles, provide the value of the horizontal and the vertical measurements as a percentage of the direct stadia measurement.
This system is sufficiently precise for locating topographic details such as rivers, bridges, buildings, and roads when an accuracy of 1/400 is acceptable.
The stadia method of distance measurement is primarily historical for surveying purposes, as distance nowadays is mostly measured by electronic or taping methods. Total station instruments do not have stadia lines marked on the reticle. Traditional methods are still used in areas where modern instruments are not common or by aficionados to antique surveying methods.
- Raymond Davis, Francis Foote, Joe Kelly, Surveying, Theory and Practice, McGraw-Hill Book Company, 1966 LC 64-66263