Area of Sectors

Area of sectors problems are the most geometry an SL student will see. The concepts are not too hard, but sometimes the visualization of geometric addition and subtraction can be challenging. Radians are also used (almost) exculisively with these types of problems. If you need a quick review go here.

### Calculating the Area of a Sector

The basic idea behind the area of a sector is that it is a portion of a circle. Knowing the area of a circle we simply have to find the portion of the circle that our sector represents. Since a full circle is $360^\circ$ or $2 \pi$ then the portion or fraction of a circle that a sector represents is:

(1)
\begin{align} \frac{\theta}{360^\circ} \end{align}

(2)
\begin{align} \frac{\theta}{2 \pi} \end{align}

The area of a circle is $A = \pi r^2$ so the area of a sector is:

(3)
\begin{align} Area = \pi r^2 \frac{\theta}{360^\circ} \end{align}

In radians this formula simplifies and results in the equation in your data booklet.

(4)
\begin{align} Area = \pi r^2 \frac{\theta}{2 \pi} = \frac {1}{2} \theta r^2 \end{align}

This equation is pretty plug-and-chug.

### Short Example(s) We have to do a bit of geometric subtraction:

(5)
\begin{equation} Area_{shaded}= Area_{sector} - Area_{triangle} \end{equation}

Using our IB data booklet:

(6)
\begin{align} Area_{shaded}= \frac{1}{2} \theta r^2 - \frac{1}{2} a b sin(\theta) \end{align}

Plug in the values and turn the crank:

(7)
\begin{align} Area_{shaded}= \frac{1}{2} \frac{\pi}{4} 3^2 - \frac{1}{2} 3^2 sin \frac{\pi}{4} \end{align}

Making sure your GDC is in the correct mode this gives an answer rounded to 3 significant figures of $Area_{shaded}=11.0$