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 ft^{2}. Therefore,

Reduce ^{120}/ _{360} to ⅓.