The supplementary units for plane angle and solid angle, historically recognized within the International System of Units (SI), are the radian and the steradian, respectively.
Historically, the International System of Units (SI) included a distinct class of "supplementary units" alongside its base and derived units. This category specifically comprised the units used to measure plane angles and solid angles.
Plane Angle: The Radian
A plane angle measures the opening between two intersecting lines or surfaces. Its supplementary unit was the radian (symbol: rad).
- Definition: A plane angle was defined as a ratio of two quantities having the same dimension of length. Specifically, one radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle. This makes the radian a dimensionless unit, as it's a ratio of length to length.
- Practical Use: Radians are fundamental in mathematics, physics, and engineering, especially in calculations involving rotational motion, oscillations, and wave phenomena. They simplify many trigonometric and calculus formulas compared to using degrees.
- For example, the formula for arc length, s = rθ, where θ is the angle in radians, becomes elegantly simple, directly relating arc length to radius.
Solid Angle: The Steradian
A solid angle is the three-dimensional equivalent of a plane angle, representing the extent of a cone-shaped region in space. Its supplementary unit was the steradian (symbol: sr).
- Definition: A solid angle was defined as a ratio of an area to the square of a length. One steradian is the solid angle subtended at the center of a sphere by a portion of the surface whose area is equal to the square of the sphere's radius. Like the radian, the steradian is a dimensionless unit, being a ratio of area to area (length² / length²).
- Practical Use: Steradians are crucial in various scientific and engineering disciplines, including:
- Optics: Used to quantify luminous intensity and flux, where light output is measured in candelas per steradian.
- Antenna Design: Describing the directional properties and radiation patterns of antennas.
- Radiation Physics: Measuring the intensity and distribution of radiation sources.
Summary of Supplementary Units
The table below summarizes these units and their historical definitions:
| Quantity | Unit | Symbol | Historical Definition |
|---|---|---|---|
| Plane Angle | Radian | rad | Ratio of two quantities having the same dimension of length (e.g., arc length to radius) |
| Solid Angle | Steradian | sr | Ratio of an area to the square of a length (e.g., surface area to the square of the radius) |
Modern Classification in SI
While historically designated as "supplementary units," the radian and steradian are now classified within the modern International System of Units (SI) as dimensionless derived units. This reclassification occurred in 1995, acknowledging that their definitions, being ratios of quantities with the same dimension, make their fundamental dimension unity. Despite this change in classification, their definitions and practical applications remain consistent. You can learn more about SI units on the official site of the Bureau International des Poids et Mesures (BIPM).
