Radians To Degrees Formula

The measure of an angle is done by the amount of counterclockwise rotation of an arm of a watch from its initial point to the terminal point. Also, when the object moves from point ‘P’ to ‘Q’ across the circumference of the circle, as we can see in the figure below:

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For instance, if I ask you to complete a 15-minute test, as the current time is 2: 05 p.m. and you will have to submit the answer sheet at sharp 2: 20 p.m. So, after every 5 minutes if a minute hand of a watch makes a 30-degree angle, then after 15 minutes, it makes an angle of 3 * 30-degrees, i.e., 90-degrees.

Similarly, if I ask you the angle made by a watch in radians, then you must know the degrees to radians formula. So, the radian measure of the same will be $\frac{\pi }{2}$. So, do you know how we applied the radian formula here? If not, please stay on this page to understand the angle to radians formula. 

Along with the convert radians to degrees formula, you will also understand the table describing the degree to radian conversion formula for various measured angles.

Arc Measure Formula

We know that the formula to convert degrees to radians is  \[\frac {\text {π radians}}{180^{\circ}} \] and we can identify it, either by its length or by measuring an angle the arc makes with the centre of the circle. The span of the arc is the arc length and the measure of an angle that arc makes is the arc measure.

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Let’s assume that an object ‘P’ moves from point A to B,  the arc measure formula is given by:

Arc measure (\[\Theta\])  =  s/r   =   arc length/radius of the circle……(1)

Since the arc measure is \[\Theta\] and the angle of the total revolution is 360°, so the portion of the angle traced by point P from its starting point to the endpoint (A to B) in fractional form is: 

\[\Theta\]/360°, i.e., arc measure/360°. 

The portion of the circumference traced by point P is the arc length. So, to trace the entire circumference of the circle, the total displacement is 2π r and the arc length will be:

Arc length      =  Central angle (\[\Theta\])/360° * circumference

Mathematically, the equation is:

s  =  2π r (\[\Theta\]/360°).........(2)

This is the required arc length formula. This equation says:

\[ \text{Arc Length} = \frac{\text{Circum ference} \ast \text{Arc Measure}}{360^{\circ}} \]

From equation (2), we get the following relation to solving problems on arc measure formula:

or,

\[\text{Arc Measure}  = \frac{\text{Arc Length(s)}\ast 360^{\circ}}{\text{circum ference}} = \frac{s\ast 360^{\circ}}{2\pi r} \]

Radian Formula

We know that a circle makes a complete revolution of 360°, while in radians, we call it 2 π radians;  this means 180° is π radians.

If π  radians = 180°

Then,  1 radian = \[ \frac{180^{\circ}}{\text{π radians}}\]

Similarly, the degrees to radians equation will be:

If 180° = π  radians 

Then, 1° = \[\frac {\text {π radians}}{180^{\circ}} \]

This is our required degrees to radian formula.

The above radian formula leads us to convert degrees to radians formula.

Degree To Radian Conversion Formula

We know that degrees to radian equation are:

1° * \[\frac {\text {π radians}}{180^{\circ}} \] =   0.01745  Rad

Below are the steps to apply radians to degrees formula:

• Step 1: Firstly, jot down the numerical value of the angle measured in degrees.

• Step 2: Secondly, multiply this mentioned value with π/180.

• Step 3: Thirdly, simplify the expression by cancelling the common terms of the numerical term.

• Step 4: Finally, the result that you have obtained after the simplification is the angle measure in radians.

The below table shows the different angle measurements in degree converted to a radian and vice versa:

Similarly, the radian to degrees conversion will be:

Now, let’s look at another example on the convert radians to degrees formula followed by the solved example on the convert degrees to radians formula.

Formula to Convert Degrees To Radians

The method we applied to convert a negative degree into radian is likewise the method for positive degrees. Multiply the measured value of the angle in degrees by π/180.

Suppose,  - 270 degrees has to be converted into radian, then,

As per the degrees to radians formula, we have:

Radian = \[\frac {\text {π radians}}{180^{\circ}} \] x  (degrees)

Radian = (π/180) x (- 270°)

Angle in radian = - \[\frac{3 \pi}{2} \]

Now, let us solve another problem on an angle to radians formula.

Let’s assume that any point P lying in the middle of the axis of the ceiling fan makes an angle of 240° with the axis of rotation of the fan, then the same measurement in radian will be?

We can writer 240°   =  150°   +  90° 

From the degrees to radians equation, we know that 150° = \[\frac{5 \pi}{6}\] , and 90°  = \[\frac{\pi}{2}\]

Adding the values in radian: \[\frac{5 \pi}{6}+\frac{\pi}{2} = \frac{4 \pi}{3} \]

Therefore, the equivalent radian of 240° is \[\frac{4 \pi}{3}\].

Conclusion

So, from the radians to degrees formula, we conclude that an angle measured in radians can be converted to degrees by multiplying the given value with \[ \frac{180^{\circ}}{\text{π radians}}\]

Similarly, from the degrees to radians formula, we need to multiply the given angle measurement in degrees with \[\frac {\text {π radians}}{180^{\circ}} \].