# Arc Length and Sectors

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.

## Arc length

As mentioned earlier in this section, an arc can be measured either in degrees or in unit length. In Figure 1l  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 2l  = 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°

sector of a circle is a region bounded by two radii and an arc of the circle.

In Figure 3OACB 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 120360 to ⅓.