As we known that there are three types of Physical quantities – Fundamental quantities , Derived quantities and Supplementary quantities .
Supplementary quantities are basically geometrical quantities of sphere and circle . Generally we use Supplementary physical quantity for calculation on secondary level .
The two supplementary physical quantities are
( i ) Plane angle
( ii ) Solid angle
Unit of supplementary quantities are called supplementary units . The supplementary units are dimensionless units , it is formed by ration of two same type of fundamental quantities that’s why it is dimensionless in the nature . The supplementary units are classified into solid angle and plane angle .
There are 2 supplementary quantities , So there will be two supplementary Unit that is radian and steradian .
( i ) Plane angle
Angle is the ratio of two lengths namely arc length to the radius making it a dimensionless quantity. We can not consider angle as a basic quantity because it is defined with the help of another basic quantity i.e. length .
Derived quantities all have dimensions which is not the case for angles .Thus, They are called supplementary quantities . Unit of plane Angle is radian .
eg . Let’s consider there is a circle of radius ‘ r ‘ and consider a small section of perimeter of length ‘ds’ . Then angle make by perimeter on center of circle is ds/r . as shown in fig. 1
dθ = ds/r
If the arc length and radius of circle will be of same length then Value of plane Angle will be 1 radian .
In geometry , generally plane angle is named as angle and angle can be defined as the figure formed by two rays meeting at a common end point. An angle is represented by the symbol ∠ .
There are many others unit of plane Angle like degree , minute and second etc .
– 1 radian = 180/π degrees , or approximately 57.296 degrees .
– 1 degree = 60 minute and 1 minute = 60 second
Note – Plane angle is always measured in 2- D
( ii ) Solid Angle
A solid angle is denoted by ω . It made up of all the lines from a closed curve meeting at a vertex , is defined by the surface area of a sphere subtended by the lines and by the radius of that sphere .
eg. Let’s consider a small surface area of area ‘ dA ‘ . All the point of area is a same distance ‘ r ‘ from a fixed point , then solid angle of made by dA area on point will be dA/r2.
Solid angle is dA/r2 , as shown in fig. 2
The unit of solid angle is the steradian and is it dimensionless . Solid angle made by a full sphere on the center of sphere is 4π steradian .
Note – Solid angle is always measured in 3-D .