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# 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:

Finding the measure of an Arc StudySmarter original

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.

 Degrees 0 30 45 60 90 120 180 270 360 Radians 0

## 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.

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.

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.

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?

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

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Question

What is the segment area of a circle?

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

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Question

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

Chord.

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Question

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

Diameter.

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Question

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

Yes.

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Question

Is the following property correct?

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

Yes.

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Question

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

Yes.

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Question

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

Yes.

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Question

Is the following statement true?

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

Yes.

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Question

A diameter divides the circle into:

Semi-circles.

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Question

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

Yes.

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Question

Two minor arcs are congruent if:

their respective chords are congruent.

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Question

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

the first chord is a diameter.

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Question

Define the 'arc' of a circle

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

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Question

Define arc measure.

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

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Question

What is a segment length?

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

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Question

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

TRUE

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Question

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

(2.5, 0.5)

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Question

What is the segment length of a circle?

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.

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Question

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

segment CB

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Question

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

7 cm

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Question

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

9.66 cm

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Question

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

(4, 3)

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Question

What are the two components of a segment

An arc and a chord

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Question

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

FALSE

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Question

The segment length is calculated using Pythagoras' theorem.

TRUE

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Question

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

(-4, 1)

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