Angle sizes can be measured in degrees when required, but with radian measure we can also measure angles in radians.
The main noticeable difference between them at first is that a radian is larger than a degree.
However they both perform the same key task, which is to measure an angle.
Further in the page will show how to find radian measure when we know an angle size in degrees.
It’s a similar situation to how both Celsius and Fahrenheit can be used to measure temperature.
Although their scales differ slightly, they are both are still perfectly reasonable measurements to use.
But in some instances, one measurement can be more handy to use than the other, and this is also the case with radians and degrees in certain sums sometimes.
Size of a Radian
It’s the case that 1 radian is roughly equal to 57.2958….. degrees.
So 1 radian is slightly more than 57 times larger than 1 degree.
One whole radian also has an arc length on the outer edge of a circle, which happens to be the same length as the radius of the circle r.
There are 2π radians in a full circle, and π radians in a half circle.
π = 3.14… , π radians = 3.14… radians2π = 6.28… , 2π radians = 6.28… radians
So the following can be understood.
\boldsymbol{\frac{\pi}{2}} radians = 90° , π radians = 180° and 2π radians = 360°.
Degrees can often be easier to work with than radians.
But there can be times where it‘s more appropriate to use radian measure instead of degrees. Particularly in Trigonometry and Calculus.
Thus it is important to know how to switch between the two measurements and be comfortable doing so.
Below are some examples of how to convert degrees to radians, and radians to degrees.
How to Find Radian Measure,
Converting Degrees to Radians
If we have an angle θ°, then the degree size of the angle can be converted to radians in the following way:
For θ°. RADIANS = \bf{\frac{\boldsymbol{\theta}}{180}} × πExamples
1.1
Convert 30° to radians.
Solution
RADIANS = \bf{\frac{30}{180}} = \bf{\frac{1}{6}} × π = \bf{\frac{\boldsymbol{\pi}}{6}}
30° = \bf{\frac{\boldsymbol{\pi}}{6}} Radians
1.2
Convert 240° to Radians:
Solution
RADIANS = \bf{\frac{240}{180}} = \bf{\frac{4}{3}} × π = \bf{\frac{4\boldsymbol{\pi}}{3}}
240° = \bf{\frac{4\boldsymbol{\pi}}{3}} Radians
1.3
Convert 137° to Radians.
Solution
There are times when the fractions for the radians don’t work out quite so cleanly.
When this happens it’s better to use decimal form, and performing multiplication with the numerical value of π.
The converting degrees to radians process though is still the same.
137° = 2.39 Radians
NOTE
2.39 radians is NOT the same value as 2.39π radians.
In the same manner that \bf{\frac{5\boldsymbol{\pi}}{2}} radians is NOT the same as \bf{\frac{5}{2}} radians.
Because \bf{\frac{5}{2}} radians is just that, \bf{\frac{5}{2}} radians. Or 1.5 radians in decimal form.
However \bf{\frac{3\boldsymbol{\pi}}{2}} radians though, is of greater value.
\bf{\frac{3 \space \times \space \boldsymbol{\pi}}{2}} = 4.71. So
\bf{\frac{3\boldsymbol{\pi}}{2}} radians is 4.71 radians.
This is a general difference that is important to keep in mind.
Converting Radians to Degrees
Converting in the other direction, radian measure can be converted to degrees by using the following formula.
DEGREES = \bf{\frac{a\boldsymbol{\pi}}{b}} × \bf{\frac{180\degree}{\boldsymbol{\pi}}}Examples
2.1
Convert \bf{\frac{2\boldsymbol{\pi}}{3}} radians to degrees.
SolutionDEGREES:
\bf{\frac{2\boldsymbol{\pi}}{3}} × \bf{\frac{180\degree}{\boldsymbol{\pi}}} = \bf{\frac{2 \space \times \space 180\degree}{3}} = \bf{\frac{360\degree}{3}} = 120°
\bf{\frac{2\boldsymbol{\pi}}{3}} Radians = 120°
2.2
Convert \bf{\frac{\boldsymbol{\pi}}{6}} radians to degrees.
SolutionDEGREES:
\bf{\frac{\boldsymbol{\pi}}{6}} × \bf{\frac{180\degree}{\boldsymbol{\pi}}} = \bf{\frac{1 \space \times \space 180\degree}{6}} = \bf{\frac{180\degree}{6}} = 30°
\bf{\frac{\boldsymbol{\pi}}{6}} Radians = 30°
2.3
Convert 1.2 radians to degrees.
Solution
Like in the case of degrees, sometimes we can be given slightly less exact values to convert.
But the approach we use still works in the same way.
1.2 × \bf{\frac{180\degree}{\boldsymbol{\pi}}} = \bf{\frac{1.2 \space \times \space 180\degree}{\boldsymbol{\pi}}} = 68.75°
1.2 Radians = 68.75°
Convert Degrees to Radians Table
The table below displays some common conversions of degrees and radians that can appear in Math.
Sometimes helpful for how to find radian measure.
