Radians
Radians, like degrees, are a unit of angular measure commonly used in fields such as math and physics. 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 radian is represented by the central angle, \(\theta\), and is measured as the length, \(a\), of the arc that subtends (or forms) the angle divided by the radius, \(r\), of the circle. Radians can be calculated as such:
Exact angles in radians are normally written in terms of \(\pi\). For example, a straight angle is referred to as radians (or \(180°\)) whereas a full rotation is referred to as \(2π\) radians (or \(360°\)).
Radians can be shown visually as such:
The area that the central angle is contained in is referred to as the sector.
Conversion from Radians to Degrees
Radians can be converted into degrees and vice versa. The conversion factor used to convert radians into degrees is:
We can use this conversion factor to convert radians into degrees:
Below is a chart outlining common radian measurements and their equivalents in degrees:
| Radians | Degrees |
|---|---|
| \(\cfrac{\pi}{6}\) rads | \(30^{\circ}\) |
| \(\cfrac{\pi}{4}\) rads | \(45^{\circ}\) |
| \(\cfrac{\pi}{3}\) rads | \(60^{\circ}\) |
| \(\cfrac{\pi}{2}\) rads | \(90^{\circ}\) |
| \(\pi\) rads | \(180^{\circ}\) |
| \(2\pi\) rads | \(360^{\circ}\) |
Example
Determine the degree measure, to the nearest tenth, for the radian measure \(\cfrac{\pi}{4}\).
In order to convert radians to degrees, we can use the degrees conversion formula:
We can then substitute \(\pi/4\) for \(x\) and simplify:
\(\theta = \cfrac{\pi}{4} \times \cfrac{180^{\circ}}{\pi}\)
\(\theta = 45^{\circ}\)
Therefore, we can determine that the degree measure of \(\cfrac{\pi}{4}\) is \(\boldsymbol{45^{\circ}}\).
\(\cfrac{8\pi}{11}\)
In order to convert radians to degrees, we can use the degrees conversion formula:
We can then substitute \(8\pi/11\) for \(x\) and simplify:
\(\theta = \cfrac{8\pi}{11} \times \cfrac{180^{\circ}}{\pi}\)
\(\theta = 130.9^{\circ}\)
Therefore, we can determine that the degree measure of \(\cfrac{\pi}{4}\) is \(\boldsymbol{130.9^{\circ}}\).
\(1.92\; [\text{rads}]\)
In order to convert radians to degrees, we can use the degrees conversion formula:
We can then substitute \(3.72\) for \(x\) and simplify:
\(\theta = 3.72 \times \cfrac{180^{\circ}}{\pi}\)
\(\theta = 213.1^{\circ}\)
Therefore, we can determine that the degree measure of \(3.72 \; \text{rads}\) is \(\boldsymbol{213.1^{\circ}}\).
Conversion from Degrees to Radians
The conversion factor used to convert degrees into radians is:
We can use this formula to convert degrees into radians:
Example
Determine an exact and an appropriate radian measure, to the nearest hundredth, for an angle of \(30°\).
In order to convert degrees to radians, we can use the radian conversion formula:
We can then substitute \(30°\) for \(x\) and simplify:
\(\theta = 30° \times \cfrac{\pi}{180^{\circ}}\)
\(\theta = \cfrac{\pi}{6}\)
In order to determine the approximate radian measure, we can use a calculator:
Therefore, for \(30°\), we can determine that its exact radian measure is \(\boldsymbol{\cfrac{\pi}{6}}\) and its approximate radian measure is \(\boldsymbol{0.52} \; [\textbf{rads}]\).
\(265.3^{\circ}\)
In order to convert degrees to radians, we can use the radian conversion formula:
We can then substitute \(30°\) for \(x\) and simplify:
\(\theta = 265.3° \times \cfrac{\pi}{180^{\circ}}\)
\(\theta = 1.473888889\pi\)
In order to determine the approximate radian measure, we can use a calculator:
Therefore, for \(265.3^{\circ}\), we can determine that its approximate radian measure is \(\boldsymbol{4.63\; [\textbf{rads}]}\).
\(98.21^{\circ}\)
In order to convert degrees to radians, we can use the radian conversion formula:
We can then substitute \(30°\) for \(x\) and simplify:
\(\theta = 98.21° \times \cfrac{\pi}{180^{\circ}}\)
\(\theta = 0.545611111\pi\)
In order to determine the approximate radian measure, we can use a calculator:
Therefore, for \(98.21^{\circ}\), we can determine that its approximate radian measure is \(\boldsymbol{1.71\; [\textbf{rads}]}\).
Degrees to Radians Calculator
Select an angle value. This function will then convert that value to radians.
Radians to Degrees Calculator
Select a radian value. This function will convert that value to degrees.