information:extraction, linguistic:translation
Measuring angles

The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be equal congruent or equal in measure.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

To measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses. The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle: θ = s r r a d. {\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad}.} Conventionally, in mathematics and the SI, the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted.

The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form k / 2 π, where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians):

θ = k 2 π ⋅ s r. {\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.}

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio s / r is unaltered.