A central angle of a circle has an angle measure of 1° if it subtends an arc that is 1/360 of the circumference of the circle. This form of angle measure is quite common. Another form of angle measure that is in use is **radian measure**. If a central angle subtends an arc that is equal to the radius of the circle (Figure ), then the central angle has a measure of one radian.

** Figure 1
** Radian measure and subtended arcs

If a central angle θ of a circle with radius *r* subtends an arc of length *q* (Figure 1 ), then its radian measure is defined as

Because both *q* and *r* are in the same units, when *q* is divided by *r* in the preceding formula, the units cancel. Therefore, *radian measure is unitless*.

**Example 1:** What is the radian measure of a central angle in a circle with radius 6 m if it subtends an arc of 24 m?

(Note that if no units are listed for an angle measure, it is assumed to be in radians.)

If θ is one complete revolution, then the subtended arc is the circumference of the circle. In this case,

Because one complete revolution is 360°,

The fact that 180° is the same as π radians is extremely important. From this relationship, the following proportion can be used to convert between radian measure and degree measure:

**Example 2:** What is the degree measure of 2.4 rad?

**Example 3:** What is the radian measure of 63°?

The radian measures of many special angles follow directly from the radian‐degree relationships. Some of these are summarized in Table 1 .

The areas of sectors of a circle are directly proportional to the measures of their central angles and directly proportional to the arcs subtended by the central angles (Figure 2 ).

** Figure 2
**

**Sector area.**

**Example 4:** Find *r* given that α = 14π and θ = π/2.

**Example 5:** Find θ if *A* = 6 and *r* = 4.

**Example 6:** What is the angle measure, in radians, of the acute angle formed by the minute and hour hands of a clock at 7:15?

The hour hand moves 1/12 of a complete revolution each hour. Therefore, every 15 minutes (one quarter of an hour), the hour hand moves 1/48 of a complete revolution. Therefore, at 7:15, the hour and minute hands are 17/48 of a revolution apart.

**Example 7:** Find the area of the shaded portion of the sector of the circle shown in Figure 7.

**Figure 3
** Drawing for Example 7.

First, use the Pythagorean theorem to find the value of *a*.

The area of the triangular (unshaded) portion of the sector can be calculated using the area formula of a triangle.

It follows that

Therefore,

area of shaded portion = area of total section − area of unshaded portion

area of shaded portion =

area of shaded portion ≈ 22.11.