Question

In the circle below, segment AB is a diameter. if the length of are ACB is 6, what is the length of the radius of the circle?

Answer 1

If AB is the diameter, it means the arc ACB is a semicircle (arc of 180°).

So, to calculate the radius of the circle, we can use the following rule of three:

$\begin{gathered} \text{arc}\to\text{length} \\ 360\degree\to2\pi r \\ 180\degree\to x \\ \frac{360}{180}=\frac{2\pi r}{x} \\ 2=\frac{2\pi r}{x} \\ 2x=2\pi r \\ x=\pi r \end{gathered}$

The length of a semicircle is given by πr. If this arc measures 6π, we have:

$\begin{gathered} \pi r=6\pi \\ \frac{\pi r}{\pi}=\frac{6\pi}{\pi} \\ r=6 \end{gathered}$

So the radius of this circle is equal 6 units.