Measuring Angles In Degrees And Radians
Understand radian measure and its relationship with degree measure for Queensland Year 11 Mathematical Methods (QCAA). A radian measures an angle by arc length, so a half turn equals pi radians.
You will learn to convert exactly between the two units, keep pi in your answers, work with signed rotations, and find coterminal angles — the foundation for the trigonometry and calculus topics ahead in the QCAA course.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 1), an angle can be measured in degrees or in radians. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius, and a half-turn gives the key link \(180^\circ=\pi\text{ rad}\). This page shows how to convert between degrees and radians, handle signed rotations, and find coterminal angles.
A radian is defined from the circle itself: if an arc has the same length as the radius, the angle it subtends at the centre is exactly one radian. Because the full circumference is \(2\pi r\), a whole turn is \(2\pi\) radians, so \(360^\circ=2\pi\text{ rad}\) and a half turn gives \(180^\circ=\pi\text{ rad}\).
To change units we use this half-turn link. To go from degrees to radians, multiply by \(\dfrac{\pi}{180}\); to go from radians to degrees, multiply by \(\dfrac{180}{\pi}\). Rotations measured anticlockwise are positive and clockwise are negative.
Two angles are coterminal when they finish on the same terminal side; you reach one from the other by adding or subtracting whole turns (\(2\pi\) or \(360^\circ\)).
The half-turn identity ties the two measures together:
Converting a measure:
The radian straight from the arc, and coterminal angles:
How to convert an angle and place it on the circle
- Choose the multiplier: degrees to radians use \(\times\dfrac{\pi}{180}\); radians to degrees use \(\times\dfrac{180}{\pi}\).
- Simplify: cancel the fraction to lowest terms, keeping \(\pi\) exact for a radian answer.
- Position or adjust: add or subtract whole turns (\(2\pi\) or \(360^\circ\)) to land in \([0,2\pi)\), then read off the quadrant.
Multiply the degree measure by \(\dfrac{\pi}{180}\):
| \(135^\circ\) | \(=\) | \(135\times\dfrac{\pi}{180}\) |
| \(=\) | \(\dfrac{135\pi}{180}\) |
Simplify the fraction (divide top and bottom by \(45\)):
| \(=\) | \(\dfrac{3\pi}{4}\) |
\(135^\circ=\dfrac{3\pi}{4}\) rad.
Multiply the radian measure by \(\dfrac{180}{\pi}\):
| \(\dfrac{7\pi}{6}\) | \(=\) | \(\dfrac{7\pi}{6}\times\dfrac{180}{\pi}\) |
| \(=\) | \(\dfrac{7\times 180}{6}\) |
Work out the arithmetic:
| \(=\) | \(7\times 30\) | |
| \(=\) | \(210^\circ\) |
\(\dfrac{7\pi}{6}=210^\circ\).
Convert (the negative sign means a clockwise rotation):
| \(-240^\circ\) | \(=\) | \(-240\times\dfrac{\pi}{180}\) |
| \(=\) | \(-\dfrac{240\pi}{180}\) | |
| \(=\) | \(-\dfrac{4\pi}{3}\) |
Add one whole turn \(2\pi=\dfrac{6\pi}{3}\) to reach \([0,\,2\pi)\):
| \(-\dfrac{4\pi}{3}+2\pi\) | \(=\) | \(-\dfrac{4\pi}{3}+\dfrac{6\pi}{3}\) |
| \(=\) | \(\dfrac{2\pi}{3}\) |
Since \(\dfrac{\pi}{2}<\dfrac{2\pi}{3}<\pi\), the terminal side lies in the second quadrant.
\(-240^\circ=-\dfrac{4\pi}{3}\), coterminal with \(\dfrac{2\pi}{3}\) (second quadrant).
Use the definition of a radian, \(\theta=\dfrac{\text{arc length}}{\text{radius}}\):
| \(\theta\) | \(=\) | \(\dfrac{\text{arc length}}{\text{radius}}\) |
| \(=\) | \(\dfrac{2.5\,r}{r}\) |
The radius cancels:
| \(=\) | \(2.5\text{ rad}\) |
The arc subtends \(2.5\) radians.
Common pitfalls
Frequently asked questions
How do you convert degrees to radians?
Multiply the degree measure by \(\dfrac{\pi}{180}\) and simplify, keeping \(\pi\) in the answer. For example \(135^\circ=135\times\dfrac{\pi}{180}=\dfrac{3\pi}{4}\).
How do you convert radians to degrees?
Multiply the radian measure by \(\dfrac{180}{\pi}\). For example \(\dfrac{7\pi}{6}\times\dfrac{180}{\pi}=210^\circ\).
What exactly is one radian?
It is the angle at the centre of a circle subtended by an arc whose length equals the radius. Since the circumference is \(2\pi r\), a full turn is \(2\pi\) radians.
Why is 180 degrees equal to pi radians?
A half turn is half the circumference, an arc of length \(\pi r\); dividing by the radius gives \(\pi\) radians, and a half turn is also \(180^\circ\).
What is a coterminal angle?
An angle that finishes on the same terminal side. Add or subtract whole turns (\(2\pi\) or \(360^\circ\)) to move between coterminal angles, for example \(-\dfrac{4\pi}{3}\) and \(\dfrac{2\pi}{3}\).