What is a Radian?

What is a Radian?

Pi movie

Roger: Dad, today our teacher told us about \( \pi \).

Dad: That's wonderful. Do you know the value of \( \pi \)?

Roger: No, Dad.

Dad: It's 3.14 Radian up to two decimal digits.

Roger: Dad, what is Radian?

Dad: Radian is a unit used to measure angles.

Roger: Thank you, Dad.

Come, let us learn in detail about the radian formula, the arc length formula, angle, the circumference, the radius, and others in this mini-lesson.

Lesson Plan

What Is the Definition of Radian?

The radian is an S.I. unit that is used to measure angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

A single radian is just below 57.3 degrees.

1 Radian image

What Is The Formula of Radian?

The angle subtended by the half-circle is \( 180^{\circ} \) which is equal to \( \pi \text { radian}\).

The angle subtended by one complete counterclockwise revolution is equal to \( 2 \pi \text { radian}\).

Pi Radian relationship

The radian formula in terms of \( \pi \) is:

\( \text{1 Radian} = \dfrac { \text {\( 180^{\circ} \)}}{ \text {\( \pi \)}} \)

To convert an angle given in degrees to radians, we just need to multiply the angle by \( \dfrac {\pi} {180}\).

\( \text{Angle in Radian} = \dfrac { \text {\( \pi \)}}{ \text {\( 180^{\circ} \)}} \text{(Angle in degrees)}\)

To convert an angle given in radians to degrees, we just need to multiply the angle by \( \dfrac {180} {\pi}\).

\( \text{Angle in Degree} =  \dfrac {\text {\( 180^{\circ} \)}} {\pi} \text{(Angle in Radian)}\)

The Arc length formula in terms of \(\theta\) and \(Radius\) is:

\[ \begin{align*} \theta &= \dfrac { \text {arc}}{ \text { radius}} \\ or \ \, \text {arc} &= \theta \times \text {radius } \end{align*} \]

It should be noted that when we use the formula  to calculate the length of the arc or the radius, we should first convert the angle \(\theta\) (if given in degrees) to radian.

How To Use the Radian Formula?

We use the radian formula because it is a dimensionless unit which is convenient and it makes calculations of the length of an arc easier.
For example, let's find the length of the arc when \(\theta\) = \( 150^{\circ} \) and \(radius = 36 \) inches
So first let's see how we convert \( 150^{\circ} \) into radians.
To convert degrees into radian, we multiply \(150^{\circ} \)  by \( \dfrac {\pi} {180}\).
\[ 150^{\circ} =  \dfrac {\pi} {180} \times 150 \]
Now,  \[ \begin{align*} \text {length of the arc } &= \theta \times r \\  \implies  \, \,  \, \, \, \, &= 150 \times \dfrac {\pi}{180} \times 36 \\ &= 30\pi\ Rad \end{align*}\]

Degree Radian
30\( ^{\circ} \) \( \dfrac { \pi}{6} \)
45\( ^{\circ} \) \( \dfrac { \pi}{4} \)
60\( ^{\circ} \) \( \dfrac { \pi}{3} \)
90\( ^{\circ} \) \( \dfrac { \pi}{2} \)
180\( ^{\circ} \) \(\pi \)
270\( ^{\circ} \) \( \dfrac { 3 \pi}{2} \)
360\( ^{\circ} \) \( 2 \pi \)

Radian Calculator

If the value of an angle is given in degrees, you can calculate its value in radian by using this calculator.

tips and tricks
Tips and Tricks


  • You can convert degrees to radians by multiplying it with \( \dfrac {\pi} {180}\).
  • You can also convert radians to degrees by multiplying it with \( \dfrac {180}{\pi} \).
  • While calculating the length of an arc, the angle (if given in degrees) should first be converted to radian.

Solved Examples

Example 1



David wants to know the arc length of a circle whose radius is 6 inches and the angle subtended by the arc is 1.5 radians. Can you help her with this?


Given, r =  6 in and \( \theta \)= 1.5 radian

\[ \begin{align*} \theta &= \dfrac { \text {arc}}{ \text { radius}} \\ or \ \, \text {arc} &= \theta \times \text {radius } \end{align*} \]

\( \implies \)\[ \text  {arc} = 1.5 \times 6 = 9 \text{ in}\]

\(\therefore\) Arc Length = 9 in

Example 2



A pendulum of length 18 inches oscillates at an angle of 42 degrees. Find the length of the arc that it covers.



Given, length of the pendulum(radius) = 18 in

Before finding the length of the arc, we need to convert the angle which is given in degrees to radians.

Since, \[ \begin{align*} \text {length of the arc } &= \theta \times r \\  \implies  \, \,  \, \, \, \, &= 42 \times \dfrac {\pi}{180} \times 18 \\ &= 4.2\pi\end{align*}\]

\(\therefore\) Length of the arc = \(4.2 \pi \) Rad
Challenge your math skills
Challenging Question
  • Verify that the value of \( \pi \) is constant by constructing two circles of radius 4 inches and 6 inches respectively?
    (Hint: Find the ratio of circumference and diameter.)

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.


Let's Summarize

This mini-lesson targeted the fascinating concept of radians. The math journey around radian starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Now, you can easily convert degree into radian and vice versa. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

FAQs on Radian

1. What is the radians to degrees formula?

\( \text{Angle in Radian} = \dfrac { \text {\( \pi \)}}{ \text {\( 180^{\circ} \)}} \text{(Angle in degrees)}\)

2. How do you simplify radian?

You can convert radian into degrees by multiplying it with \( \dfrac {180}{\pi} \).

3. What is the measure of 0 degrees in radian?

0 degrees is equal to 0 radians.

4. What is 2 \(\pi\) called?

2\( \pi \) is also called Tau.

5. What exactly is a radian?

Radian is the SI unit of measuring angles based on the arc length and the radius.

6. How many radians are in a circle?

There are 2 \(\pi\) radians in a circle.

7. What is radian and degree?

Radian and degrees are the units in which angles are measured.

8. What is arc length formula?

\( \text{Arc length} = \dfrac { \text {2 \(\pi\)  r }}{ \text {\( 360^{\circ} \)}} \times \theta\)

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