# What is a Radian?

**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.

**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 \( 2 \pi \text { radian}\).

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.

**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?

**Solution**

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

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

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

\(\therefore\) |

Example 2 |

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

**Solution**

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 |

- 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\)