Angular distance can be used to calculate height and distances in trigonometry.
Angular distance is a measure of the apparent separation between two points from the perspective of the observer. Straight lines that extend from each point to the intersection of the observer. The angle at which these two lines intersect is the angular distance and is usually expressed in degrees or radians. In trigonometry, this angle can be used to calculate heights and distances. Astronomers often use the angle to describe the apparent separation between celestial bodies without referring to their actual distance.
A common trigonometry problem involves calculating the height of a building. The angular separation of the line of sight between the top and bottom of the building at a known distance is enough information to determine its height. Calculations involving angular distance are common in surveying and aiming. Instead of degrees or radians, military personnel often find it useful to express target calculations in terms of angular mils. This is 1/6400 of the circumference of a circle, or more conveniently, the angular distance between two points one meter apart in an interval of 1000 meters.
In astronomy, there are two basic ways to describe the position of an object in the sky. One is by reference to a coordinate system, the other is by the position of the object relative to another body. In the equatorial coordinate system, the poles of the Earth and the equator are projected into space as the celestial poles and the celestial equator. The position of a body is described by its declination, degrees north or south of the celestial equator, and its hour angle. This is the angular distance along the celestial equator between the observer’s location and the celestial meridian, a circle that passes directly above the observer and through the celestial poles.
For the hobbyist, angular distance can be used to help locate an astronomical object relative to a known body, or simply to note an interesting feature. Often all that is needed is an outstretched hand. With the arm extended, the tip of the little finger subtends about one degree of arc. Three middle fingers subtend about four degrees and a clenched fist about ten. The distance from the little finger to the thumb of an open hand covers about 18 degrees.
The more serious professional observer often uses a similar measure of angular distance called the angular diameter. This is the apparent size of an astronomical object as seen from Earth. These diameters are very small and are usually measured in seconds of arc, or 1/3600th of a degree. As with ground measurement, if the distance of an object is known, its angular diameter can be used to calculate its actual size.