Select your language

Suggested languages for you:
Log In Start studying!
StudySmarter - The all-in-one study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
|
|

All-in-one learning app

  • Flashcards
  • NotesNotes
  • ExplanationsExplanations
  • Study Planner
  • Textbook solutions
Start studying

Arc Measures

Arc Measures

It is very important to be familiar with the anatomy of a circle and especially the angles within it. This article covers the properties of arc measures, the formula for an arc measure, and how to find it within a geometric context.

The arc and its measure

There are two important definitions to be aware of:

The arc of a circle

An arc is the edge of a circle sector, i.e. the edge bounded/delimited by two points in the circle.

Arc length is the size of the arc, i.e. the distance between the two delimiting points on the circle.

The measure of an arc

If we think of an arc as being the edge between two points A and B on a circle, the arc measure is the size of the angle between A, the centre of the circle, and B.

In relation to the arc length, the arc measure is the size of the angle from which the arc length subtends.

Here are these definitions demonstrated graphically:

Arc measures finding the measure of an arc StudySmarterFinding the measure of an Arc StudySmarter original

Radians versus degrees

Before we introduce the formula for arc measurement, let’s recap degrees and radians.

To convert radians to degrees: divide by and multiply by 180.

To convert degrees to radians: divide by 180 and multiply by.

Here are some of the common angles which you should recognise.

Degrees030456090120180270360
Radians0

Arc measure and arc length formulae

Finding the arc measure with the radius

The formula that links both the arc measure (or angle measure) and the arc length is as follows:

Where

  • r is the radius of the circle
  • is the arc measure in radians
  • S is the arc length

We can find the arc measure given the radius and the arc length by rearranging the formula: .

Find the arc measure shown in the following circle in terms of its radius, r.

Arc measures Arc measures StudySmarter

Using the formula :

We need the arc measure in terms of r, so we need to rearrange this equation:

Finding the arc measure with the circumference

If we are not given the radius, r, then there is a second method for finding the arc measure. If we know the circumference of a circle as well as the arc length, then the ratio between the arc measure and (or depending on whether you want the arc measure in degrees or radians) is equal to the ratio between the arc length and the circumference.

Where

  • c is the circumference of the circle

  • is the arc measure in degrees
  • S is the arc length

Find the arc length, x, of the following circle with a circumference of 10 cm.

Arc measures finding x StudySmarter

Using the formula :

Rearranging, we get:

to 3 s.f.

Arc Measures - Key takeaways

  • An arc is the edge of a circle sector, i.e. the edge bounded/delimited by two points in the circle.
  • Arc length is the size of the arc, i.e. the distance between the two delimiting points on the circle.
  • An arc measure is the size of the angle from which the arc subtends.
  • Finding the arc measure given the radius and arc length:
    • Where

      • r is the radius of the circle.

      • is the arc measure in radians.
      • S is the arc length.

  • Finding the arc measure given the circumference and arc length:

    • Where:

      • c is the circumference of the circle.

      • is the arc measure in degrees.
      • S is the arc length.

Frequently Asked Questions about Arc Measures

An arc measure is the angle from which an arc of a circle subtends.

How to find the measure of an arc: given the radius and arc length, the arc measure is the arc length divided by the radius. Given the circumference, the ratio between the arc measure and 360 degrees is equal to the ratio between the arc length and the circumference.

The arc measure is the arc length divided by the radius. 

The arc measure is the arc length divided by the radius. 

In geometry, the arc measure is the arc length divided by the radius. 

Final Arc Measures Quiz

Question

What is segment length?

Show answer

Answer

Segment length is the distance between two points on straight line.

Show question

Question

What is the segment area of a circle?

Show answer

Answer

It is the area bound by a chord and the circle's edge.

Show question

Question

A line segment that has its endpoints on a circle is called?

Show answer

Answer

Chord. 

Show question

Question

A chord passing through the center of the circle is called?

Show answer

Answer

Diameter. 

Show question

Question

Does a chord, apart from a diameter, any other chord splits a circle into a major arc and a minor arc?


Show answer

Answer

Yes. 

Show question

Question

Is the following property correct?


Equal chords of a circle subtend equal angles at the center. 

Show answer

Answer

Yes. 

Show question

Question

Does a perpendicular bisector from the centre of a circle bisects a chord into equal halves?

Show answer

Answer

Yes. 

Show question

Question

Are chords that are equidistant from the center of the circle equal in lengths?

Show answer

Answer

Yes.

Show question

Question

Is the following statement true?


The intersecting chords theorem states that the products of the intercepts on intersecting chords are equal. 

Show answer

Answer

Yes. 

Show question

Question

A diameter divides the circle into: 

Show answer

Answer

Semi-circles. 

Show question

Question

Will the corresponding arc lengths be equal if the chords are of equal lengths?

Show answer

Answer

Yes. 

Show question

Question

Two minor arcs are congruent if:

Show answer

Answer

their respective chords are congruent. 

Show question

Question

If one chord is a bisector of another chord, then:

Show answer

Answer

the first chord is a diameter. 

Show question

Question

Define the 'arc' of a circle

Show answer

Answer

An arc, or arc length, is the edge of a circle sector.

Show question

Question

Define arc measure.

Show answer

Answer

In relation to the arc length, the arc measure is the angle from which the arc length subtends.

Show question

Question

What is a segment length?

Show answer

Answer

The segment length is the distance between two points on a line segment.

Show question

Question

The lengt of segment can be determined using the coordinates of two points.

Show answer

Answer

TRUE

Show question

Question

The midpoint G between points A, (2, 4) and B, (3, -3) is...

Show answer

Answer

(2.5, 0.5)

Show question

Question

What is the segment length of a circle?

Show answer

Answer

The line segment of a circle can either be the diameter of a circle when the line passes through the center of the circle or a chord if the line passes any other place apart from the center of a circle.

Show question

Question

The segment length between points C and B would be called...

Show answer

Answer

segment CB

Show question

Question

Find the length of the line segment of a circle with a radius of 7 cm which subtends 60° at the center.

Show answer

Answer

7 cm

Show question

Question

Find the length of the line segment of a circle with a radius of 5 cm which subtends 210° at the center.

Show answer

Answer

9.66 cm

Show question

Question

Find the midpointd between the origin and point Z (8, 6)

Show answer

Answer

(4, 3)

Show question

Question

What are the two components of a segment

Show answer

Answer

An arc and a chord

Show question

Question

The segment length cannot be calculated when the endpoint and midpoint are given.

Show answer

Answer

FALSE

Show question

Question

The segment length is calculated using Pythagoras' theorem.

Show answer

Answer

TRUE

Show question

Question

The midpoint between a certain pont J (2, 5) and another point W is (-1, 3). Find the coordinates for point W.

Show answer

Answer

(-4, 1)

Show question

60%

of the users don't pass the Arc Measures quiz! Will you pass the quiz?

Start Quiz

Discover the right content for your subjects

No need to cheat if you have everything you need to succeed! Packed into one app!

Study Plan

Be perfectly prepared on time with an individual plan.

Quizzes

Test your knowledge with gamified quizzes.

Flashcards

Create and find flashcards in record time.

Notes

Create beautiful notes faster than ever before.

Study Sets

Have all your study materials in one place.

Documents

Upload unlimited documents and save them online.

Study Analytics

Identify your study strength and weaknesses.

Weekly Goals

Set individual study goals and earn points reaching them.

Smart Reminders

Stop procrastinating with our study reminders.

Rewards

Earn points, unlock badges and level up while studying.

Magic Marker

Create flashcards in notes completely automatically.

Smart Formatting

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.