Radians

Radians: It is another unit used to measure angles.
1 radian is defined as the angle subtended by an arc such that its arc length is equal to the radius of the circle.
\[{\rm{Arc\;Length}} = {\rm{Radius}} \;\;\;\Rightarrow\;\;\; {\rm{Angle}} = 1\;rad\]

The angle subtended by an arc is 2 radians when its arc length is 2 times the radius of the circle.
\[{\rm{Arc\;Length}} = 2 \times {\rm{Radius}} \;\;\;\Rightarrow\;\;\; {\rm{Angle}} = 2\;rad\]
The angle subtended by an arc is 3 radians when its arc length is 3 times the radius of the circle.
\[{\rm{Arc\;Length}} = 3 \times {\rm{Radius}} \;\;\;\Rightarrow\;\;\; {\rm{Angle}} = 3\;rad\]
The angle subtended by an arc is \(n\) radians when its arc length is \(n\) times the radius of the circle.
\[{\rm{Arc\;Length}} = n \times {\rm{Radius}} \;\;\;\Rightarrow\;\;\; {\rm{Angle}} = n\;rad\]
The angle (\(180^\circ\)) subtended by an arc is \(\pi\) radians when its arc length is \(\pi\) times the radius of the circle (i.e. semicircle).
\[{\rm{Arc\;Length}} = \underbrace {\pi \times {\rm{Radius}}}_{{\rm{Semicircle}}} \;\;\;\Rightarrow\;\;\; {\rm{Angle}} = \underbrace {\pi\;rad}_{180^\circ}\]
The angle (\(360^\circ\)) subtended by an arc is \(2\pi\) radians when its arc length is \(2\pi\) times the radius of the circle (i.e. circumference).
\[{\rm{Arc\;Length}} = \underbrace {2\pi \times {\rm{Radius}}}_{{\rm{Circumference}}} \;\;\;\Rightarrow\;\;\; {\rm{Angle}} = \underbrace {2\pi\;rad}_{360^\circ}\]
Hence,

$$\begin{align}
180^\circ & = \pi {\rm{\;}}rad \\
360^\circ & = 2\pi {\rm{\;}}rad \\
\end{align}$$



Below shows a circle, with centre O.
The points A, B, P, Q, R, S, T, U and X are points on the circumference of the circle.

You can:

• shift the point O to change the position of the circle
• shift the point X to change the size of the circle
• shift the points A and B to change their positions and the arc length of AB

Instructions:

1. Adjust the arc length of AB.
2. Check all the boxes.
3. Shift the point B to coincide with the point P such that the arc length, AB is equal to the radius.
4. Observe the values of \(\frac{{{\rm{Arc\;Length}}}}{{{\rm{Radius}}}}\) and ∠AOB.
5. Shift the point B to coincide with the point Q such that the arc length, AB is 2 \(\times\) the radius.
6. Observe the values of \(\frac{{{\rm{Arc\;Length}}}}{{{\rm{Radius}}}}\) and ∠AOB.
7. Shift the point B to coincide with the point R such that the arc length, AB is 3 \(\times\) the radius.
8. Observe the values of \(\frac{{{\rm{Arc\;Length}}}}{{{\rm{Radius}}}}\) and ∠AOB.
9. Shift the point B to coincide with the point S such that the arc length, AB is 4 \(\times\) the radius.
10. Observe the values of \(\frac{{{\rm{Arc\;Length}}}}{{{\rm{Radius}}}}\) and ∠AOB.
11. Shift the point B to coincide with the point T such that the arc length, AB is 5 \(\times\) the radius.
12. Observe the values of \(\frac{{{\rm{Arc\;Length}}}}{{{\rm{Radius}}}}\) and ∠AOB.
13. Shift the point B to coincide with the point U such that the arc length, AB is 6 \(\times\) the radius.
14. Observe the values of \(\frac{{{\rm{Arc\;Length}}}}{{{\rm{Radius}}}}\) and ∠AOB.
15. Repeat steps 1 ~ 14 for other radii of the circle.

Note: To see the animation, click on the "play" button (at bottom left).



Download the GeoGebra file: http://www.geogebratube.org/material/download/format/file/id/47082

Download the Worksheet: http://db.tt/YykgIMXY