Turn (geometry) in the context of Ratio

In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or ⁠${\displaystyle \pi }$/2⁠ radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.

Closely related and important geometrical concepts are perpendicular lines, meaning lines that form right angles at their point of intersection, and orthogonality, which is the property of forming right angles, usually applied to vectors. The presence of a right angle in a triangle is the defining factor for right triangles, making the right angle basic to trigonometry.

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a unit of measurement of a plane angle in which one full rotation is assigned the value of 360 degrees.

It is not an SI unit—the SI unit of angular measure is the radian—but it is mentioned in the SI brochure as an accepted unit. Because a full rotation equals 2π radians, one degree is equivalent to ⁠π/180⁠ radians.

A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol ′, is a unit of angular measurement equal to ⁠1/60⁠ of a degree. Since one degree is ⁠1/360⁠ of a turn, or complete rotation, one arcminute is ⁠1/21600⁠ of a turn. The nautical mile (nmi) was originally defined as the arc length of a minute of latitude on a spherical Earth, so the actual Earth's circumference is very near 21600 nmi. A minute of arc is ⁠π/10800⁠ of a radian.

A second of arc, arcsecond (abbreviated as arcsec), or arc second, denoted by the symbol ″, is a unit of angular measurement equal to ⁠1/60⁠ of a minute of arc, ⁠1/3600⁠ of a degree, ⁠1/1296000⁠ of a turn, and ⁠π/648000⁠ (about ⁠1/206264.8⁠) of a radian.

Two-dimensional rotation can occur in two possible directions or senses of rotation. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands relative to the observer: from the top to the right, then down and then to the left, and back up to the top. The opposite sense of rotation or revolution is (in Commonwealth English) anticlockwise (ACW) or (in North American English) counterclockwise (CCW). Three-dimensional rotation can have similarly defined senses when considering the corresponding angular velocity vector.

In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function ${\displaystyle F}$ of some real variable ${\displaystyle t}$ (such as time) is an angle-like quantity representing the fraction of the cycle covered up to ${\displaystyle t}$. It is expressed in such a scale that it varies by one full turn as the variable ${\displaystyle t}$ goes through each period (and ${\displaystyle F(t)}$ goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or ${\displaystyle 2\pi }$ as the variable ${\displaystyle t}$ completes a full period.

This convention is especially appropriate for a sinusoidal function, since its value at any argument ${\displaystyle t}$ then can be expressed as ${\displaystyle \varphi (t)}$, the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.)

In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. It is a circular arc that measures 180° (equivalently, π radians, or a half-turn). It only has one line of symmetry (reflection symmetry).

In non-technical usage, the term "semicircle" is sometimes used to refer to either a closed curve that also includes the diameter segment from one end of the arc to the other or to the half-disk, which is a two-dimensional geometric region that further includes all the interior points.

The angular displacement (symbol θ, ϑ, or φ) – also called angle of rotation, rotational displacement, or rotary displacement – of a physical body is the angle (with the unit radian, degree, turn, etc.) through which the body rotates (revolves or spins) around a centre or axis of rotation. Angular displacement may be signed, indicating the sense of rotation (e.g., clockwise); it may also be greater (in absolute value) than a full turn.

In a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, π radians, two right angles, or a half-turn). A triangle has three angles, and has one at each vertex, bounded by a pair of adjacent sides.

The sum can be computed directly using the definition of angle based on the dot product and trigonometric identities, or more quickly by reducing to the two-dimensional case and using Euler's identity.

In trigonometry, the gradian – also known as the gon (from Ancient Greek γωνία (gōnía) 'angle'), grad, or grade – is a unit of measurement of an angle, defined as one-hundredth of the right angle; in other words, 100 gradians is equal to 90 degrees. It is equivalent to ⁠1/400⁠ of a turn, ⁠9/10⁠ of a degree, or ⁠π/200⁠ of a radian. Measuring angles in gradians (gons) is said to employ the centesimal system of angular measurement, initiated as part of metrication and decimalisation efforts.

In continental Europe, the French word centigrade, also known as centesimal minute of arc, was in use for one hundredth of a grade; similarly, the centesimal second of arc was defined as one hundredth of a centesimal arc-minute, analogous to decimal time and the sexagesimal minutes and seconds of arc. The chance of confusion was one reason for the adoption of the term Celsius to replace centigrade as the name of the temperature scale.

A clock face is the part of an analog clock (or watch) that displays time through the use of a flat dial with reference marks, and revolving pointers turning on concentric shafts at the center, called hands. In its most basic, globally recognized form, the periphery of the dial is numbered 1 through 12 indicating the hours in a 12-hour cycle, and a short hour hand makes two revolutions in a day. A long minute hand makes one revolution every hour. The face may also include a second hand, which makes one revolution per minute. The term is less commonly used for the time display on digital clocks and watches.

A second type of clock face is the 24-hour analog dial, widely used in military and other organizations that use 24-hour time. This is similar to the 12-hour dial above, except it has hours numbered 1–24 (or 0–23) around the outside, and the hour hand makes only one revolution per day. Some special-purpose clocks, such as timers and sporting event clocks, are designed for measuring periods less than one hour. Clocks can indicate the hour with Roman numerals or Hindu–Arabic numerals, or with non-numeric indicator marks. The two numbering systems have also been used in combination, with the prior indicating the hour and the latter the minute. Longcase clocks (grandfather clocks) typically use Roman numerals for the hours. Clocks using only Arabic numerals first began to appear in the mid-18th century.