Measure of an Arc

Measure of an Arc

How to find the measure of an arc: definition, examples, and their solutions.

Definition

The measure of an arc is the measure of the arc's central angle.

The measure of an arc
is the measure of the arc's central angle.

Example 1

Find the measure of arc AB. [measure of angle AB] = 60, [measure of angle CD] = 45.

The central angle of [arc AB], ∠AOB,
is 60º.

So m[arc AB] = 60.

When it says [arc AB],
it usually means the minor arc.

(whose measure is less than 180,
smaller than the half circle)

Example 2

Find the measure of arc ADB. [measure of angle AB] = 60, [measure of angle CD] = 45.

[arc ADB] is the blue arc.

This is the way to say the major arc.

(whose measure is greater than 180,
bigger than the half circle)

[arc ADB] = [circle] - [arc AB].

So m[arc ADB] = 360 - m[arc AB].

The central angle of [arc AB], ∠[brown],
is 60º.

So m[arc ADB] = 360 - 60.

360 - 60 = 300

So m[arc ADB] = 300.

Example 3

Find the measure of arc DE. [measure of angle AB] = 60, [measure of angle CD] = 45.

These two blue angles are the vertical angles.

So these two angles are congruent.

So m∠DOE = 60.

Vertical angles

The central angle of [arc DE], ∠DOE,
is 60º.

So m[arc DE] = 60.

Example 4

Find the measure of arc BC. [measure of angle AB] = 60, [measure of angle CD] = 45.

∠[blue], ∠[green], and ∠[brown]
form a line: AD.

m∠[blue] = [60]
m∠[brown] = [45]

So [60] + m∠[green] + [45] = 180.

60 + 45 = 105

Move +105 to the right side.

180 - 105 = 75

So m∠[green] = 75.

The central angle of [arc BC], ∠[green],
is 75º.

So m[arc BC] = 75.