Angle Conversions
Popular Comparisons
| ° | rad | grad | arcmin | arcsec | |
|---|---|---|---|---|---|
| 1 degree | 1 | 0.01745 | 1.111 | 60 | 3,600 |
| 1 radian | 57.296 | 1 | 63.662 | 3,437.75 | 206,265 |
| Right angle | 90 | 1.5708 | 100 | 5,400 | 324,000 |
| Half turn | 180 | 3.1416 | 200 | 10,800 | 648,000 |
| Full circle | 360 | 6.2832 | 400 | 21,600 | 1,296,000 |
Angle
An angle is a geometric measure of the rotation between two rays sharing a common endpoint, or the inclination between two intersecting lines. The two principal units for measuring angles are degrees (°) and radians (rad). The degree, dividing a full circle into 360 equal parts, originates from ancient Babylonian astronomy, which used base-60 arithmetic and approximated the year as 360 days. The radian, the SI-coherent unit, is defined as the angle subtended at the centre of a circle by an arc equal in length to the radius, making a full rotation exactly 2π radians (≈ 6.2832 rad).
Radians are preferred in mathematics and physics because they simplify formulas dramatically. The derivative of sin(x) equals cos(x) only when x is expressed in radians; in degrees, a correction factor of π/180 must be applied. Similarly, small-angle approximations — fundamental in optics, pendulum mechanics, and structural engineering — are only valid when angles are expressed in radians. The formula for arc length (s = rθ), angular velocity (ω = θ/t), and rotational kinetic energy all require radians for dimensional consistency.
The gradian (also called gon), which divides a full circle into 400 equal parts, was introduced during the French metric system reforms and is still used in land surveying, civil engineering, and military mapping in parts of Europe. Arcminutes (1/60 of a degree) and arcseconds (1/3600 of a degree) are used in astronomy, GPS coordinates, and optical instrument specifications — the Hubble Space Telescope, for example, has an angular resolution of approximately 0.05 arcseconds.