From the time of the ancient Babylonians, it has been the practice to divide circles into $360$360 small arcs. The angle subtended at the centre by any one of those arcs is called one degree. In effect, the angle, in degrees, an arc subtends, is equivalent to the fraction of 360 defined by the ratio of the arc length $l$l to the overall circumference of a circle with radius $r$r. That is:
| $\theta$θ | $=$= | $\frac{l}{2\pi r}\times360^\circ$l2πr×360° |
Radians
In a similar way, we could define the angle subtended by an arc at the centre of a circle, by the ratio of the arc length to the radius. That is:
| $\theta$θ | $=$= | $\frac{l}{r}$lr |
Angles defined this way are called radians and an angle in radians can be calculated as $\theta=\frac{l}{r}$θ=lr, where $l$l is the arc length and $r$r is the radius of the circle. Hence, the angle subtended by an arc whose length is equal to the radius is $1$1 radian.
Radians are an alternate way to describe angles and are the international standard unit for measuring angles. This radian measure of angles simplifies calculations, as seen with the equations above, in current and future areas of our mathematics learning.
Because angles in radian measure are just a ratio, they do not require a unit, although some writers indicate that radian measure is being used by adding a superscript c after a number or the abbreviation rad, like this: $2^c$2c or $2$2$rad$rad. (The c is short for circular-measure.)
We will now restrict our attention to angles in the unit circle. We measure angles subtended at the centre by arcs of this circle. One radian is the angle that an arc of one unit subtends at the centre of a circle of radius 1 unit.
Let's consider the ratio of the circumference of the unit circle to the radius as this is the radian measure of a complete revolution:
| $\theta$θ | $=$= | $\frac{2\pi}{1}$2π1 |
| $\theta$θ | $=$= | $2\pi$2π |
Hence, the angle, in radians, represented by a full turn around the circle is $2\pi$2π. This is equivalent to $360^\circ$360°.
Note that the radian measure works for all circles, we use the unit circle to simplify the development of the concept. For example, for a circle with radius $2$2, a full revolution in radians would be:
| $\theta$θ | $=$= | $\frac{2\pi2}{2}$2π22 |
| $\theta$θ | $=$= | $2\pi$2π |
Common angles and conversions
A half-circle makes an angle of $\pi$π radians or $180^\circ$180° and a right-angle is $\frac{\pi}{2}$π2radians.
An angle of $1$1 radian must be:
$\frac{360^\circ}{2\pi}=\frac{180^\circ}{\pi}\approx57.3^\circ$360°2π=180°π≈57.3°
In mathematical notation: $1^c=\frac{180}{\pi}.$1c=180π.We use this equation to help us convert radians to degrees.
To convert from radians to degrees, we can rearrange the equation above to get $1^\circ=\frac{\pi}{180}$1°=π180.
In practice, angles given in radian measure are usually expressed as fractions of $\pi$π.
$2\pi$2π radians = $360^\circ$360°
$\pi$π radians = $180^\circ$180°
$1^\circ$1° = $\frac{\pi}{180}$π180 radians
To convert from degrees to radians: multiply by $\frac{\pi}{180}$π180
To convert from radians to degrees: multiply by $\frac{180}{\pi}$180π
Practice questions
QUESTION 1
Convert $90^\circ$90° to radians.
Give your answer in exact form.
QUESTION 2
Convert $-300^\circ$−300° to radians.
Give your answer in exact form.
QUESTION 3
Convert $\frac{2\pi}{3}$2π3 radians to degrees.
QUESTION 4
Convert $4.2$4.2 radians to degrees.
Give your answer correct to one decimal place.
Question 5
Given that $x=\pi$x=πc represents half a circle, what fraction of the circumference of the unit circle does $x=\frac{4\pi}{7}$x=4π7c represent?