1.7 Supplementary Units in the SI System

In the International System of Units (SI), physical quantities are described using a set of fundamental (or base) units and derived units. Historically, a third category known as "supplementary units" existed to define geometrical quantities, specifically plane and solid angles. While this classification has been officially updated, the units themselves remain essential in science and engineering.

Prerequisites

• An understanding of fundamental units (e.g., meter, kilogram, second).
• Familiarity with derived units, which are combinations of base units (e.g., newton, joule). For more on derived units, see Derived Units→.

Definition and Current Status

Previously, supplementary units were a distinct class in the SI system. However, in 1995, the International Committee for Weights and Measures decided to eliminate this category. The two supplementary units, the radian and the steradian, were reclassified as dimensionless derived units.

This means they are now formally considered to be derived from the base units, but they are dimensionless because they are ratios of two similar quantities (length/length or area/area). For more on dimensional analysis, see Dimensions→.

The SI Supplementary Units

Despite their reclassification, the radian and steradian are unique and crucial for measuring angles.

1. Radian (rad)

Physical Quantity: Plane Angle

Definition: One radian is the angle at the center of a circle subtended by an arc that is equal in length to the radius of the circle.

Relationship to Degrees: A full circle contains $2\pi$ radians, which is equivalent to 360 degrees. Therefore, 1 radian is approximately 57.3 degrees.

Formula: The angle in radians ($\theta$) is the ratio of the arc length ($s$) to the radius ($r$): $\theta =\frac{s}{r}$

2. Steradian (sr)

Physical Quantity: Solid Angle (three-dimensional angle)

Definition: One steradian is the solid angle subtended at the center of a sphere by a portion of the surface that has an area equal to the square of the sphere's radius.

Relationship to a Full Sphere: A complete sphere subtends a solid angle of $4\pi$ steradians at its center.

Formula: The solid angle in steradians ($\Omega$) is the ratio of the subtended surface area ($A$) to the square of the radius ($r^2$): $\Omega =\frac{A}{r^2}$

Examples

1. Radian Example: In a circle with a radius of 5 meters, an arc with a length of 5 meters will subtend an angle of exactly 1 radian at the center.

2. Steradian Example: On a sphere with a radius of 3 meters, a patch of the surface with an area of 9 m² (which is $3^2$ m²) will subtend a solid angle of 1 steradian at the sphere's center.