By definition, a sector of a circle is the region bounded by two radii of the circle and their intercepted arc.

Our mission here is to determine how to find the area of a sector of the circle. Notice that if the area of the entire circle is

A_{C},

then to find the area of a sector we need to figure out how much of the circle is represented by the sector.

To find the area of each sector shown above, we multiply the area of the entire circle by the portion of it represented by the sector.

S_{1} = \frac{1}{4} \cdot A_{C} S_{2} = \frac{2}{3} \cdot A_{C} S_{3} = \frac{7}{8} \cdot A_{C}

Next, we will find the measure of the intercepted arcs of each of the sectors considered above.

Using the angles labeled before, let's make the following table.

Central Angle of Sector $1$	Central Angle of Sector $2$	Central Angle of Sector $3$
$9 0^{\circ} \frac{9 0 ^{\circ}}{3 6 0 ^{\circ}} = \frac{1}{4} \cdot 36 0^{\circ} = \frac{1}{4}$	$24 0^{\circ} \frac{2 4 0 ^{\circ}}{3 6 0 ^{\circ}} = \frac{2}{3} \cdot 36 0^{\circ} = \frac{2}{3}$	$31 5^{\circ} \frac{3 1 5 ^{\circ}}{3 6 0 ^{\circ}} = \frac{7}{8} \cdot 36 0^{\circ} = \frac{7}{8}$

As we can see, to find the portion of a circle represented by a sector we divide the measure of the intercepted arc by

36 0^{\circ} .

Therefore, by using the expression written below we can find the area of a sector of a circle.

Area of sector A C B = \frac{m A B}{3 6 0 ^{\circ}} \cdot A_{C}

Extra

Finding the Area of a Circle

Let's divide a circle with radius $r$ into congruent sections.

Next, let's rearrange all the sections to form a figure that approximates a parallelogram. Notice that increasing the number of congruent sections makes our figure look more like a parallelogram.

As we can see, the base of the figure above approaches half of the circumference — that is, $b = \frac{1}{2} (2 π r) = π r .$ Also, the height of our figure is equal to the radius: $h = r .$ Let's find the area of our figure.

Since our figure was made from the circle, we can conclude that both have the same area. Thus, the area of a circle is equal to $π r^{2} .$

Exercise