If you know the area of the entire circle, what would be the area of one-fourth of the circle? Find the measure of the central angle and divide it by 360^(∘) to know the portion of the circle represented by the sector.
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 = 1/4* A_C
S_2 = 2/3* A_C
S_3 = 7/8* 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.
As we can see, to find the portion of a circle represented by a sector we divide the measure of the intercepted arc by 360^(∘). Therefore, by using the expression written below we can find the area of a sector of a circle.
Area of sector ACB = mAB/360^(∘) * 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= 12(2π 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.
