**Arc Measure**

**Meaning of Arc Measure**

Arcs are measured in degrees. This is not the same as the *length* of an arc, which depends on the size of the circle.

The measure of a minor arc is defined as the measure of its central angle:

The measure of a major arc is 360^{o} minus the measure of the corresponding minor arc:

**Lengths of Arcs**

The measure of an arc is given in degrees. The *length* of an arc depends on the size of the circle and is measured in standard units, such as inches, centimeters, etc. The circumference of a circle is an arc measuring 360^{o}. The length of the circumference is given by the formula: *C* = *πd*, where *d* is the diameter of the circle. This formula can also be given as: *C* = 2*πr*, where *r* is the radius. An arc measuring *θ* degrees is *θ* /360 of the circumference, so its length can be found by either of these formulas:

or

Example 1: Find the length of a 72^{o} arc in a circle of radius 10 cm.

Solution:

length = 72/360 Χ 20*π* = 4*π* cm or about 12.57 cm

**Area of a Sector**

A sector of a circle is the region enclosed by an arc and two radii to its endpoints:

Since the area of a circle is given by *A* = *π r*^{2}, the area of a sector whose arc measures *θ* degrees is given by the formula:

**A "Slice"**

Example 2: Find the area of the shaded slice:

Solution: The area of the shaded region can be found by subtracting the area of the triangle from the area of the sector:

The triangle consists of two 30-60-90 triangles, so its sides and altitude can be determined:

The area of the sector is 120/360 times 36*π* cm^{2}, or about 37.7 cm^{2}. The area of the triangle is:

So the area of the desired slice is about 37.7 – 15.6 = 22.1 cm^{2}.

**Inscribed Angles**

An inscribed angle is an angle with its vertex on the circle and whose sides are chords of the circle:

When a side of an inscribed angle contains a diameter of the circle, it is easy to see that the measure of the inscribed angle is half the measure of its intercepted arc:

because triangle *AOB* is isosceles (*BO* and *AO* are radii). But

because an external angle of a triangle is the sum to the two remote interior angles. Therefore . But the measure of arc *AC* is the same as the measure of the central angle, *AOC*.

**The Measure of an Inscribed Angle**

In general, the measure of the inscribed angle is half the measure of its intercepted arc:

Angle *ABD* measures half of arc *AD* and angle *DBC* measures half of arc *DC*. Therefore angle *ABC* measures half of arc *ADC*.

We state this as a theorem:

Theorem 3: The measure of an inscribed angle is half the measure of its intercepted arc:

**Two Inscribed Angles with the Same Arc**

An important consequence of this is:

Theorem 4: If two inscribed angles incept the same arc, then they are congruent.

Since angles *ABD* and *ACD* both intercept arc *AD* each measures half that arc, so they have the same measure and are therefore congruent.

Example 3: What is the measure of arc *ABC*?

Solution: Since angle *ADC* is an inscribed angle, it is half its intercepted arc. Therefore the measure of that arc is twice this angle, or 160^{o}.

**Arcs of Intersecting Chords**

If two chords intersect, their vertical angles are congruent, and each vertical angle is the *average* of the two intercepted arcs:

We can prove this result by joining endpoints of the chords to form a triangle with *q* as an exterior angle:

*θ* is an exterior angle of triangle *CEP*, so it is the sum of the two remote interior angles, angles *ECD* and *CEF*. But these are inscribed angles with intercepted arcs whose measures are *a* and *b*, so each measures half its arc:

**Arcs Intercepted by Secants**

In the following figure, *PT* and *QT* are secants to circle *O*. The measure of angle *T* is related to the measures of arcs *PQ* and *RS* by the given formula:

The proof of this is similar to the proof that the measure of the angle formed by two intersecting chords is the average of intercepted arcs.

In triangle *QRT*, angle *PRQ* is an exterior angle, so it is equal to the sum of its remote interior angles:

But angle *PRQ* is an inscribed angle, so its measure is half of its intercepted arc, of half of *a*. Likewise, angle *RQT* is also an inscribed angle, so it measures half of *b*. Substituting these in the above equation gives us:

Therefore,

**Arcs Intercepted by Tangents and Secants**

If one of the secant lines is moved away from the center of the circle until it becomes a tangent line, the same result holds:

The same is true about two tangents with a common external point:

In fact, since *a* + *b* = 360^{o}, this last result can also be written as:

or as

**Return to Lesson 5**