Students are often confused by the fact that the arcs of a circle are capable of being measured in more than one way. The best way to avoid that confusion is to remember that arcs possess two properties. They have length as a portion of the circumference, but they also have a measurable curvature, based upon the corresponding central angle.
As mentioned earlier in this section, an arc can be measured either in degrees or in unit length. In Figure 1, l is a connected portion of the circumference of the circle.
Figure 1 Determining arc length.
The portion is determined by the size of its corresponding central angle. A proportion will be created that compares a portion of the circle to the whole circle first in degree measure and then in unit length.
With the use of this proportion, l can now be found. In Figure 1, the measure of the central angle = 120°, circumference = 2π r, and r = 6 inches.
Reduce 120°/360° to ⅓.
Example 1: In Figure 2, l = 8π inches. The radius of the circle is 16 inches. Find m ∠ AOB.
Reduce 8π/32π to ¼.
Figure 2 Using the arc length and the radius to find the measure of the associated central angle.
So, m ∠ AOB = 90°
A sector of a circle is a region bounded by two radii and an arc of the circle.
In Figure 3, OACB is a sector. is the arc of sector OACB. OADB is also a sector. is the arc of sector OADB. The area of a sector is a portion of the entire area of the circle. This can be expressed as a proportion.
Figure 3 A sector of a circle.
Example 2: In Figure 4, find the area of sector OACB.
Figure 4 Finding the area of a sector of a circle.
Example 3: In Figure 5, find the area of sector RQTS.
Figure 5 Finding the area of a sector of a circle.
The radius of this circle is 36 ft, so the area of the circle is π(36)2 or 1296π ft2. Therefore,
Reduce 120/ 360 to ⅓.