An angle (from Latin angulus, from Greek ἀγκύλος (ankýlos), derived from the Indo-European root ank, to bend, to curve), in mathematics, is each of the two portions of a plane between two line segments having the same origin (vertex). It may also be called a plane angle to distinguish it from the derived concept of a solid angle. The half-lines are called the sides of the angle, and their origin is called the vertex of the angle. The term, thus defined, refers to concepts that are widely used, especially in geometry and trigonometry.

Each angle is associated with an amplitude, the measure related to the position of one half of the ray with respect to the other, and thus to the conformation of the part of the plane forming the angle: it is expressed in sexagesimal degrees, in sexadecimal degrees, in centesimal degrees, or in radians, always with real values.

The association of the angle with a direction introduces angle amplitudes with signs, which allow the definition of trigonometric functions with even negative real arguments. Signed amplitudes provide essential contributions to the possibilities of calculus and applications to classical physics and related quantitative disciplines.

Convex and Concave Angle

A concave angle is the angle that contains the extensions of the half-lines (sides) that form it. A convex angle is the part of the plane that does not contain the extensions of the half-lines that divide the plane. Convex angles have amplitudes between 0 and 180 sexagesimal degrees, 0 to 200 centesimal degrees, 0 to π radians; while concave angles have amplitudes between 180 to 360 degrees, 200 to 400 centesimal degrees, π to 2π radians. The amplitudes are always non-negative.

If the half-lines are different but belong to the same line, each of the two half-planes defined by the line with the vertex (which distinguishes the half-lines) is called a flat angle.

Angle amplitude measurement systems

In the sexagesimal system, the complete angle, or round angle, is divided into 360 segments, corresponding to the conventional unit of measurement called the sexagesimal degree, denoted by the symbol °. The reason for the division of the round angle into 360 segments can be traced back to the astronomical use of this measurement by the Babylonians: since the Sun makes one complete revolution on the celestial sphere in the course of one year, which was then estimated to be about 360 days, one degree roughly corresponds to the Sun’s displacement on the ecliptic in one day.

The name “sexagesimal degree” derives from the fact that the subunits of the degree, the minute and the second, are divided into sixtieths; thus, as in the clock, each degree is divided into 60 minute primes, denoted by the symbol ‘ and called simply minutes, and each minute is divided into 60 minute seconds, denoted by the symbol ‘ and called simply seconds. Further subdivisions of the second follow the common decimal system instead. This subdivision is due to the fact that a sexagesimal numbering system was in vogue in ancient Babylonia, which has come down to us as a historical legacy in the clock and on protractors.

The amplitude of an angle could thus be expressed in a form such as 57° 17′ 44.8″.

Over time, other systems of measurement were introduced to make it easier to measure the amplitude of an angle. At the end of the eighteenth century, even the sexagesimal system was subject to attempts at rationalization: a centesimal system was proposed, based precisely on the centesimal degree as the hundredth part of the right angle, chosen as the fundamental angle to replace 90 with the rounder and more convenient 100, although it did not find practical use until around 1850, when Ignazio Porro used it to construct his first centesimal division instruments. In this system, the round angle is divided into 400 equal segments with decimal fractional submultiples. It is still a conventional unit of measurement, not motivated by any mathematical reason.

Since the development of calculus, another unit of measurement, in some respects more “motivated” or “natural”, has gained more and more importance: the radian, understood as the ratio of the length of an arc of a circle to the radius of the circle itself, since this ratio does not depend on the radius, but only on the angle involved. Thus, the round angle measures 2π, which is the ratio of the length of the circumference to its radius.

To sum up, to measure the amplitude of the angle, the most attested systems of measurement are

  • the centesimal system, where the unit is the centesimal degree;
  • the sexagesimal system, with the sexagesimal degree as its unit;
  • the sexadecimal system, whose unit is the sexadecimal degree. It is a variant of the former, with the round angle divided into 360 parts, and the submultiples of the degrees expressed in decimal form;
  • the radian system, or mathematical system, with the unit of measurement being the radian.

In military usage, the thousandth of a radian, commonly called the “thousandth,” is used to determine deviations and related corrections in artillery shots. On a circumference with a radius of one kilometer, it is equivalent to a string one meter long. For example, to correct a shot fired 100 meters to the right of a target placed at a distance of 10 km, a correction of 10°° (thousandths) of red is required. The graduated scale observed inside some binoculars is expressed in thousandths of radians, the red color means rotation to the left, the green color means rotation to the right.

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