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Measurement [ edit ] Since the angular distance (or separation) is conceptually identical to an angle, it is measured in the same units , such as degrees or radians , using instruments such as goniometers or optical instruments specially designed to point in well-defined directions and record the corresponding angles (such as telescopes ).
In navigation the instrument is also called a cross-staff and was used to determine angles, for instance the angle between the horizon and Polaris or the sun to determine a vessel's latitude, or the angle between the top and bottom of an object to determine the distance to said object if its height is known, or the height of the object if its distance is known, or the horizontal angle between ...
This task can be, for example, to triangulate a scale map on site using a plane table drawing of intersecting lines in the direction of the object from two or more points or to measure the angle and horizontal distance to the object from some reference point's polar measurement. Angles measured can be horizontal, vertical or in any chosen plane.
Abney levels typically include scales graduated in measure degrees of arc, percent grade, and in topographic Abney levels, grade in feet per surveyor's chain, and chainage correction. The latter is the cosine of the angle, used to convert distances measured along the slope to horizontal distances. By using trigonometry the user of an Abney ...
The interior angle concept can be extended in a consistent way to crossed polygons such as star polygons by using the concept of directed angles.In general, the interior angle sum in degrees of any closed polygon, including crossed (self-intersecting) ones, is then given by 180(n–2k)°, where n is the number of vertices, and the strictly positive integer k is the number of total (360 ...
The measure of ∠AOB, where O is the center of the circle, is 2α. The inscribed angle theorem states that an angle θ inscribed in a circle is half of the central angle 2θ that intercepts the same arc on the circle. Therefore, the angle does not change as its vertex is moved to different positions on the circle.
Since angle B is supplementary to both angles C and D, either of these angle measures may be used to determine the measure of Angle B. Using the measure of either angle C or angle D, we find the measure of angle B to be 180° − (180° − x) = 180° − 180° + x = x. Therefore, both angle A and angle B have measures equal to x and are equal ...
Angle trisection is the construction, using only a straightedge and a compass, of an angle that is one-third of a given arbitrary angle. This is impossible in the general case. For example, the angle 2 π /5 radians (72° = 360°/5) can be trisected, but the angle of π /3 radians (60°) cannot be trisected. [8]