Arc: Measure

How to find the measure of an arc: definition, 3 examples, and their solutions.

Definition

Arc

An arc is the piece of a circle.

The given arc is read as [arc AB].

Measure of an Arc

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

(θ is read as [theta].)

Example 1

Example

Solution

Arc AB is this blue arc.

Just like arc AB,
if the arc is smaller than the half circle,
then it's called a minor arc.

For most cases,
arc AB means the minor arc (= the smaller arc).

The measure of arc AB is
the measure of the central angle:
60.

So 60 is the answer.

Example 2

Example

Solution

Arc APB is this blue arc.

By following the letters A, P, and B,
you'll get this blue arc.

Just like arc APB,
if the arc is bigger than the half circle,
then it's called a major arc.

The measure of arc AB is
the measure of the central angle.

Set the central angle of the arc θ.

θ and 60º form 360º.

So θ + 60 = 360.

Move +60 to the right side.

Then θ = 300.

So 300 is the answer.

Example 3

Example

Solution

Arc BC is this green arc.

The measure of arc BC is
the measure of the central angle.

Set the central angle of the arc θ.

AD passes through the center of the circle.

So 60º, θ, and 45º
are all central angles.

And 60º, θ, and 45º
form a line (= diameter).

So 60 + θ + 45 = 180.

60 + 45 = 105

Move +105 to the right side.

Then θ = 75.

So 75 is the answer.