# What are Supplementary Units?

Supplementary units are considered revolutionary units in the field of measurement that brought the revolution in angular measurements. Students often find it difficult to gather information about questions like What are the supplementary units? How radian was originated? How many radians and steradians account for the circumference of the circle and sphere respectively? Check out this article to make your concepts clear.

Supplementary units are the pure geometrical units that are used to measure the angles i-e Plane angles and Solid Angles.

## Numbers of Supplementary Units:

** **The supplementary units are two in the numbers.

- Radian
- Steradian

The details of these two supplementary units are discussed below.

## Radian (A supplementary Unit)

** **Radian is the angle made by an arc whose length is equal to the same circle’s radius.

### Explanation** **

Radian is the measurement unit of plane angle (angle in two dimensions). Consider a circle having radius ‘r’ in which an arc of length ‘s’ is taken such that s=r. Then according to the definition of a radian, the angle made by an arc at the center of a circle will be equal to one radian

**Further Explore: Relation Between Radian and Degree**

### Radians account for the circumference of circle

The number of radians in general is given by the arc length divided by the radius of a circle.

\text{Number of radian }(\theta) = \frac{\text{Arc length}}{\text{radius}}

\text{Number of radians = }\frac{s}{r} \text{~~~~~here s= Arc length}

\textbf{For one revolution } s=2\pi r

\text{Number of radian = }\frac{2\pi r}{r}

\text{Number of radian = } 2\pi

From the above derivation, it is clear that the **radians account for 2 π rad (6.28) for the circumference of the circle.**

## Steradian

Steradian is the supplementary unit that can be stated as the angle formed at the center of a sphere by an area of its surface equal to the square of the radius of that sphere.

### Explanation

Steradian is the unit of solid angle (angle in three dimensions). It is denoted by sr. Consider a sphere of radius ‘r’ and the surface area of the sphere is equal to 4*πr^*2If we take out a small portion of the surface area of the sphere such that A=r^2. Then according to the definition, it is equal to one steradian.

**Explore More: Examples Of Radian And Degree Relation**

### Steradians account for the circumference of a sphere

**B****y definition of Steradian**

The solid angle subtended by a closed sphere is given by the area of the sphere divided by the square of the radius.

\text{Number of steradians in one revolution } = \frac{\text{Area of sphere}}{\text{Square of radius}}

\textbf{As we know that}

\text{Area of sphere} = 4\pi r^2

\textbf{putting value of area of sphere}

\text{Number of steradians in one revolution }= \frac{ 4\pi r^2}{r^2}

\text{Number of steradians in one revolution }= 4\pi

## Some Common and Important Questions Regarding Radian

While studying the radian, several questions arise in an individual mind. Such as What is meant by radian in supplementary units? There are how many radians in the circumference of the circle? Some of these common and important questions for clearing the concept of radian are given below.

### How Radian was Originated?

In ancient times there was a single unit (Degree) for the calculation and measurement of angles, but in 1714 Roger Cotes (English Mathematician) gave the concept of Radian. In 1722, Robert Smith (Cousin of Roger Cotes) collected the Calculation of cotes and gave the relation between one radian and degree. Smith calculated one radian in terms of degrees and distinguished the radian as a unit for the measurement of angles. This new unit was then adopted by Leonhard Euler in 1765 particularly to define angular velocity.

Initially, the newly originated unit (Radian) was commonly known as the **Circular measure of an angle.** The term radian was first printed on 5^{th} June 1873 in the questions of an examination, put by James Thomson.

### What do you know about radian as SI Unit?

Radian as a SI unit has created great controversy and confusion among scientists. In 1960, the **CGPM (General Conference of Weight and Measures) **classified the radian and steradian as supplementary units but they were unable to clarify that either these supplementary units (radian and steradian) are base units or derived units.

In May 1980, **CCU (Consultative Committee for Units) **proposed to classify the radian as a base unit based on a constant alpha knot equal to 1 radian. The CGPM (In October 1980) decided to allow the freedom to use supplementary units as derived units or non-derived units. Later on, the CGPM stated radian and steradian as **dimensionless derived units **instead of supplementary units.

### What are the uses of radian?

Radian is in extensive use in various fields such as mathematics, physics, prefixes, and variants. The use of radian in these different areas is discussed below.

#### In Physics

The radian plays a significant role in physics. Units of various angular quantities such as angular velocity, angular acceleration, phase, and phase difference are expressed in terms of radians.

For example, angular velocity is expressed in **radian per second (rad/s)**. Similarly, the unit often used for angular acceleration is **radian per second square (rad/s ^{2})**. Also, phase and phase differences are expressed in terms of radian.

#### In Mathematics

Radian is very advantageous in various branches of mathematics. Most of the angle measurements are expressed in terms of radians as it gives an elegant result. Besides it, radian is also very useful in trigonometric functions.

**Explore Further: What is 60 Degrees in Radians**

#### In Prefixes

**R**adian is also very useful in prefixes to make the measurement easier. Some of the radian prefixes are milliradian and microradian etc.

## Frequently Asked Questions – FAQs

**Q#1 When radian is invented?**

Radian was invented in 1714 but was first printed on 5^{th} June 1873 in an examination paper.

**Q#2 What are the dimensions of supplementary units?**

Supplementary units are dimensionless quantities.

**Q#3 How many radians account for the circumference of a circle?**

In the circumference of a circle, there are 2π (6.28) radians

### **Q#4 How many steradians account for the circumference of a sphere?**

There are 4π (12.56) steradians at any of the anterior points of the circumference of a sphere.

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