Radians

In the applet above, the \(x\)-coordinates of the blue, green and orange points are given by the angle. As can be seen by the labels on the \(x\)-axis, one revolution does not correspond to 360 but to \(2\pi\). In other words, one revolution corresponds to the total path travelled by the red point, which is the circumference of the unit circle.

You can change the angle unit to radians in GeoGebra under Options->Settings..., then choose the Advanced tab.

To see why using degrees as unit is inappropriate, see the applet below. In that applet, degrees are used. In order to see the shape of the graphs, choose the tool Move Graphics View , then drag the \(x\)-axis until \(360^\circ\) can be seen. By doing so, the unit circle is deformed into an ellipse.

Exercises

Exercise 1 - Arcs

• Make a unit circle through two points \(A=(0,0)\) and \(B=(1,0)\).
• Enter a point \(C\) on the circle and mark the angle \(\angle BAC\).
• Make sure that radians is chosen under the Advanced-tab under Options->Settings....
• Use the tool Circular Arc with Centre between Two Points on the points \(A, B, C\).
• Change the radius of the circle and the angle. Observe the length of the arc.

What is the length of the arc in terms of the radius \(r\) and the angle \(\alpha\), if radians are used as angle unit?

What is the length if degrees are used as angle unit?

Exercise 2 - Sectors

• Delete the arc.
• Use the tool Circular Sector with Centre between Two Points on the points \(A, B, C\).
• Change the radius of the circle and the angle. Observe the area of the sector.

What is the area of the sector in terms of the radius \(r\) and the angle \(\alpha\), if radians are used as angle unit?

What is the area if degrees are used as angle unit?