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# How to find arc length

An arc can come from a central angle, which is one whose vertex is located at the center of the circle. You can measure an arc in two different ways:

As an angle. The measure of an arc as an angle is the same as the central angle that intercepts it.

As a length. The length of an arc is directly proportional to the circumference of the circle and is dependent on both the central angle and the radius of the circle.

with r representing the radius. Also recall that a circle has 360 degrees. So if you need to find the length of an arc, you need to figure out what part of the whole circumference (or what fraction) you’re looking at.

You use the following formula to calculate the arc length: The symbol theta (θ) represents the measure of the angle in degrees, and s represents arc length, as shown in the figure:

The variables involved in computing arc length.

If the given angle theta is in radians,

Time for an example. To find the length of an arc with an angle measurement of 40 degrees if the circle has a radius of 10, use the following steps:

Assign variable names to the values in the problem.

The angle measurement here is 40 degrees, which is theta. The radius is 10, which is r.

Plug the known values into the formula.

This step gives you

Simplify to solve the formula.

which multiplies to

The arc length for an angle measurement of 40 degrees

and a radius of 16. This time, you must solve for theta (the formula is s = rθ when dealing with radians):

To use the arc length calculator, simply enter the central angle and the radius into the top two boxes. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it’s not too difficult to calculate.

The calculator will then determine the length of the arc . It will also calculate the area of the sector with that same central angle.

## How to Calculate the Area of a Sector and the Length of an Arc

Our calculators are very handy, but we can find the arc length and the sector area manually. It’s good practice to make sure you know how to calculate these measurements on your own.

### How to Find the Arc Length

An arc length is just a fraction of the circumference of the entire circle. So we need to find the fraction of the circle made by the central angle we know, then find the circumference of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters.

First, let’s find the fraction of the circle’s circumference our arc length is. The whole circle is 360°. Let’s say our part is 72°. We make a fraction by placing the part over the whole and we get $$\frac<72><360>$$ , which reduces to $$\frac<1><5>$$ . So, our arc length will be one fifth of the total circumference. Now we just need to find that circumference.

The circumference can be found by the formula C = πd when we know the diameter and C = 2πr when we know the radius, as we do here. Plugging our radius of 3 into the formula, we get C = 6π meters or approximately 18.8495559 m.

Now we multiply that by $$\frac<1><5>$$ (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Note that our units will always be a length.

### How to Find the Sector Area

Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a fraction of the area of the circle. So to find the sector area, we need to find the fraction of the circle made by the central angle we know, then find the area of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters.

First, let’s find the fraction of the circle’s area our sector takes up. The whole circle is 360°. Our part is 72°. We make a fraction by placing the part over the whole and we get $$\frac<72><360>$$ , which reduces to $$\frac<1><5>$$ . So, our sector area will be one fifth of the total area of the circle. Now we just need to find that area.

The area can be found by the formula A = πr 2 . Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m 2 .

Now we multiply that by $$\frac<1><5>$$ (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared. Note that our answer will always be an area so the units will always be squared.

After the radius and diameter, another important part of a circle is an arc. In this article, we will discuss what an arc is, find the length of an arc, and measure an arc length in radians. We will also study the minor arc and major arc.

## What is an Arc of a Circle?

An arc of a circle is any portion of the circumference of a circle. To recall, the circumference of a circle is the perimeter or distance around a circle. Therefore, we can say that the circumference of a circle is the full arc of the circle itself.

## How to Find the Length of an Arc?

The formula for calculating the arc states that:

Arc length = 2πr (θ/360)

Where r = the radius of the circle,

θ = the angle (in degrees) subtended by an arc at the center of the circle.

360 = the angle of one complete rotation.

From the above illustration, the length of the arc (drawn in red) is the distance from point A to point B.

Let’s work out a few example problems about the length of an arc:

Example 1

Given that arc, AB subtends an angle of 40 degrees to the center of a circle whose radius is 7 cm. Calculate the length of arc AB.

The length of an arc = 2πr(θ/360)

Length = 2 x 3.14 x 7 x 40/360

Example 2

Find the length of an arc of a circle that subtends an angle of 120 degrees to the center of a circle with 24 cm.

The length of an arc = 2πr(θ/360)

= 2 x 3.14 x 24 x 120/360

Example 3

The length of an arc is 35 m. If the radius of the circle is 14 m, find the angle subtended by the arc.

The length of an arc = 2πr(θ/360)

35 m = 2 x 3.14 x 14 x (θ/360)

Multiply both sides by 360 to remove the fraction.

Divide both sides by 87.92

θ = 143.3 degrees.

Example 4

Find the radius of an arc that is 156 cm in length and subtends an angle of 150 degrees to the circle’s center.

The length of an arc = 2πr(θ/360)

156 cm = 2 x 3.14 x r x 150/360

Divide both sides by 2.6167

So, the radius of the arc is 59.62 cm.

## How to Find the Arc Length in Radians?

There is a relationship between the angle subtended by an arc in radians and the ratio of the length of the arc to the radius of the circle. In this case,

θ = (the length of an arc) / (the radius of the circle).

Therefore, the length of the arc in radians is given by,

S = r θ

where, θ = angle subtended by an arc in radians

S = length of the arc.

r = radius of the circle.

One radian is the central angle subtended by an arc length of one radius, i.e., s = r

The radian is just another way of measuring the size of an angle. For instance, to convert angles from degrees to radians, multiply the angle (in degrees) by π/180.

Similarly, to convert radians to degrees, multiply the angle (in radians) by 180/π.

Example 5

Find the length of an arc whose radius is 10 cm and the angle subtended is 0.349 radians.

Arc length = r θ

Example 6

Find the length of an arc in radians with a radius of 10 m and an angle of 2.356 radians.

Arc length = r θ

Example 7

Find the angle subtended by an arc with a length of 10.05 mm and a radius of 8 mm.

Arc length = r θ

Divide both sides by 8.

There, the angle subtended by the arc is 1.2567 radians.

Example 8

Calculate the radius of a circle whose arc length is 144 yards and arc angle is 3.665 radians.

Arc length = r θ

Divide both sides by 3.665.

Example 9

Calculate the length of an arc which subtends an angle of 6.283 radians to the center of a circle which has a radius of 28 cm.

Arc length = r θ

Minor arc (h3)

The minor arc is an arc that subtends an angle of less than 180 degrees to the circle’s center. In other words, the minor arc measures less than a semicircle and is represented on the circle by two points. For example, arc AB in the circle below is the minor arc.

Major arc (h3)

The major arc of a circle is an arc that subtends an angle of more than 180 degrees to the circle’s center. The major arc is greater than the semi-circle and is represented by three points on a circle.

In order to fully understand Arc Length and Area in Calculus, you first have to know where all of it comes from. And that’s what this lesson is all about!

Arc Length, according to Math Open Reference, is the measure of the distance along a curved line.

In other words, it’s the distance from one point on the edge of a circle to another, or just a portion of the circumference.

So how do we calculate this, since we only know how to find distances for straight lines, not curves?

Well, it all comes down to radians and proportions!

What we discover is that the length of an arc of a circle is proportional to the measure of it’s central angle.

Arc Length Formula

All this means is that by the power of radians and proportions, the length of an arc is nothing more than the radius times the central angle!

We will use our new found skills of finding arc length to see how one wheel can turn another, as well as how many inches a pulley can lift a weight.

Additionally, we will see how right triangles and fractions can help us find the Area of a Sector, which, according to Cool Math, is the area between two segments coming out of the center of a circle.

Ok, all this means is that a sector is just a fancy way of saying a portion of a circle (i.e., slice of pizza or a slice of pie).

Again, we will use our new found formula and apply it to several problems where we are asked to find area.

These two concepts are going to be so helpful when we get to calculus, and are asked to find the arc length and area of things other than circles. Because, we will be armed with the power of circles, triangles, and radians, and will see how to use our skills and tools to some pretty amazing math problems. I can’t wait!

## Arc Length – Worksheet

: This handout contains 7 examples on finding arc length given radius and central angles.

## Finding Arc Length – Video

Find the length of an arc of a circle if the radius is and the angle is radians.

Write the formula to find the arc length given the angle in radians.

### Example Question #2 : Find The Length Of The Arc Of A Circle Using Radians

Find the length of a circular arc if the radius is , with an angle of .

Write the formula to find the length of an arc given the angle in radians.

Substitute the radius and the angle in order to find the length of the arc.

### Example Question #1 : Find The Length Of The Arc Of A Circle Using Radians

If circle has a radius of fathoms and sector has a central angle of . What is the measure of arc ?

To find the length of an arc, use the following formula:

We are given the radius, as well as our central angle, so plug in what we know and simplify

So our answer is 5 fathoms

### Example Question #4 : Find The Length Of The Arc Of A Circle Using Radians

Find the length of an arc of a circle if the radius is and the angle is .

Write the formula for arc length.

Substitute the known radius and angle.

### Example Question #5 : Find The Length Of The Arc Of A Circle Using Radians

When CERN was designing its Large Hadron Collider, they needed to run a cable from the control house to a spot radians around the circle. If the Large Hadron Colider has a radius of , what length of cable will be necessary to reach the sensor?

When CERN was designing its Large Hadron Collider, they needed to run a cable from the control house to a spot radians around the circle. If the Large Hadron Colider has a radius of , what length of cable will be necessary to reach the sensor?

The formula for length of an arc is as follows:

Thus, to find the measure of the central angle, what we are really doing is multiplying the total circumference by the fractional part of the circle we are interested in. In this case, we already have the radius and the central angle, we just need to plug and chug.

### Example Question #6 : Find The Length Of The Arc Of A Circle Using Radians

If a circle has a circumference of 310 kilometers, find the length of the arc associated with a central angle of radians.

If a circle has a circumference of 310 kilometers, find the length of the arc associated with a central angle of radians

Recall that arc length can be found via the following:

Upon closer examination, we see that the formula is really two parts. The first part gives us the fractional area of the circle we care about. The second part is simply the circumference. This means that if we multiply the whole circumference by the fraction that we care about, we will just get the arc length of the portion we care about.

So, plug it in and go!

Looks messy, but we’ll simplify it.

### Example Question #7 : Find The Length Of The Arc Of A Circle Using Radians

Find the arc length of a circle with circumference that goes from to around its center.

Remember that arc length of a circle is given by:

To get , we convert from degrees to radians.

To get the radius we convert from the circumference

### Example Question #8 : Find The Length Of The Arc Of A Circle Using Radians

If the length of is and , which of the following is closest to the length of minor arc ?

The formula for arc length of a circle, given in radians, is , where is the arc length.

Then, plug in the givens: and simplify: .

Then round to the appropriate placement: .

Therefore, the arc length of is .

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A circle is a single sided shape, but can also be described as a locus of points where each point is equidistant (the same distance) from the centre.

## Angle Formed by Two Rays Emanating from the Center of a Circle

An angle is formed when two lines or rays that are joined together at their endpoints, diverge or spread apart. Angles range from 0 to 360 degrees.
We often "borrow" letters from the Greek alphabet to use in math and science. So for instance we use the Greek letter "p" which is π (pi) and pronounced "pie" to represent the ratio of the circumference of a circle to the diameter.
We also use the Greek letter θ (theta) and pronounced "the – ta", for representing angles.

An angle is formed by two rays diverging from the centre of a circle. This angle ranges from 0 to 360 degrees

360 degrees in a full circle

## Parts of a Circle

A sector is a portion of a circular disk enclosed by two rays and an arc.
A segment is a portion of a circular disk enclosed by an arc and a chord.
A semi-circle is a special case of a segment, formed when the chord equals the length of the diameter.

Arc, sector, segment, rays and chord

## What is Pi (π) ?

Pi represented by the Greek letter π is the ratio of the circumference to the diameter of a circle. It's a non-rational number which means that it can't be expressed as a fraction in the form a/b where a and b are integers.

Pi is equal to 3.1416 rounded to 4 decimal places.

## What's the Length of the Circumference of a Circle?

If the diameter of a circle is D and the radius is R.

## Servius Tullius: The Sixth King of Ancient Rome

So in terms of the radius R

## What's the Area of a Circle?

The area of a circle is A = πR 2

So the area in terms of the radius R is

## What are Degrees and Radians?

Angles are measured in degrees, but sometimes to make the mathematics simpler and elegant it's better to use radians which is another way of denoting an angle. A radian is the angle subtended by an arc of length equal to the radius of the circle. ( "Subtended" means produced by joining two lines from the end points of the arc to the center).

An arc of length R where R is the radius of a circle, corresponds to an angle of 1 radian

So if the circumference of a circle is 2πR i.e 2π times R, the angle for a full circle will be 2π times 1 radian or 2π radians.

And 360 degrees = 2π radians

A radian is the angle subtended by an arc of length equal to the radius of a circle.

## How to Convert From Degrees to Radians

Dividing both sides by 360 gives

1 degree = 2π /360 radians

Then multiply both sides by θ

So to convert from degrees to radians, multiply by π/180

## How to Convert From Radians to Degrees

Divide both sides by 2π giving

1 radian = 360 / (2π) degrees

Multiply both sides by θ, so for an angle θ radians

So to convert radians to degrees, multiply by 180/π

## How to Find the Length of an Arc

You can work out the length of an arc by calculating what fraction the angle is of the 360 degrees for a full circle.

A full 360 degree angle has an associated arc length equal to the circumference C

So 360 degrees corresponds to an arc length C = 2πR

Divide by 360 to find the arc length for one degree:

1 degree corresponds to an arc length 2πR/360

To find the arc length for an angle θ, multiply the result above by θ:

So arc length s for an angle θ is:

The derivation is much simpler for radians:

By definition, 1 radian corresponds to an arc length R

So if the angle is θ radians, multiplying by θ gives:

Arc length is Rθ when θ is in radians

## What are Sine and Cosine?

A right-angled triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse and it is the longest side. Sine and cosine are trigonometric functions of an angle and are the ratios of the lengths of the other two sides to the hypotenuse of a right-angled triangle.

In the diagram below, one of the angles is represented by the Greek letter θ.

The side a is known as the "opposite" side and side b is the "adjacent" side to the angle θ.

### cosine θ = length of adjacent side / length of hypotenuse

Sine and cosine apply to an angle, not necessarily an angle in a triangle, so it's possible to just have two lines meeting at a point and to evaluate sine or cos for that angle. However sine and cos are derived from the sides of an imaginary right angled triangle superimposed on the lines. In the second diagram below, you can imagine a right angled triangle superimposed on the purple triangle, from which the opposite and adjacent sides and hypotenuse can be determined.

In this explainer, we will learn how to find the arc length and the perimeter of a circular sector and solve problems including real-life situations.

We can begin by recalling the terminology used to describe parts of a circle. Firstly, we remember that the arc of a circle is a section of the circle between two radii. However, given two radii, there are two possible arcs between the two radii. We can see an example of this in the following diagram.

Both arcs are a section of the circle between two given radii, so to avoid confusion, we denote the larger arc as the major arc and the smaller one as the minor arc.

This is equivalent to saying that if the central angle is less than

radians , then we know it is minor. If it is larger than these values, then it is a major arc. We can then define circular arcs as below.

### Definition: An Arc of a Circle

An arc of a circle is a section of the circumference of the circle between two radii.

Given two radii, we denote the larger of the arcs as the major arc and the smaller of the arcs as the minor arc. The larger arc is the one with the largest central angle.

If the two arcs are the same length, then we call these semicircular arcs. These occur when the central angle is

We can now see how we find the length of an arc of a circle. Let’s say we have the arc below.

We can find the length of any arc subtended by an angle by first recalling how we find the circumference of a circle, the distance around the outside of the circle.

, of a circle of radius

The length of the minor arc above can be calculated by multiplying the circumference,

gives the length of the parametrized curve whose Cartesian coordinates x i are functions of t .

interprets the x i as coordinates in the specified coordinate chart.

ArcLength is also known as length or curve length. A one-dimensional region can be embedded in any dimension greater than or equal to one. The ArcLength of a curve in Cartesian coordinates is . In a general coordinate chart, the ArcLength of a parametric curve is given by , where is the metric. In ArcLength [ x , < t , t min , t max > ] , if x is a scalar, ArcLength returns the length of the parametric curve < t , x >. Coordinate charts in the third argument of ArcLength can be specified as triples < coordsys , metric , dim >in the same way as in the first argument of CoordinateChartData . The short form in which dim is omitted may be used. The following options can be given:
•  AccuracyGoal Infinity digits of absolute accurary sought Assumptions $Assumptions assumptions to make about parameters GenerateConditions Automatic whether to generate conditions on parameters PerformanceGoal$PerformanceGoal aspects of performance to try to optimize PrecisionGoal Automatic digits of precision sought WorkingPrecision Automatic the precision used in internal computations

Symbolic limits of integration are assumed to be real and ordered. Symbolic coordinate chart parameters are assumed to be in range given by the "ParameterRangeAssumptions" property of CoordinateChartData . ArcLength can be used with symbolic regions in GeometricScene .

## Basic Examples     (3)

The length of the line connecting the points , , and :

The length of a circle with radius :

Circumference of a parameterized unit circle:

Length of one revolution of the helix , , expressed in cylindrical coordinates:

## Scope     (16)

### Special Regions     (3)

Lines can be used in any number of dimensions:

Only a 1D Simplex has meaningful arc length:

It can be embedded in any dimension:

### Formula Regions     (2)

The arc length of a circle represented as an ImplicitRegion :

The arc length of a circle represented as a ParametricRegion :

Using a rational parameterization of the circle:

### Mesh Regions     (2)

The arc length of a MeshRegion :

The arc length of a BoundaryMeshRegion in 1D:

### Derived Regions     (4)

The portion of a circle intersecting a disk:

The arc length of a Circle intersected with a Triangle :

### Parametric Formulas     (5)

An infinite curve in polar coordinates with finite length:

The length of the parabola between and :

Arc length specifying metric, coordinate system, and parameters:

Arc length of a curve in higher-dimensional Euclidean space:

The length of a meridian on the two-sphere expressed in stereographic coordinates:

## Options     (3)

### Assumptions     (1)

The length of a cardioid with arbitrary parameter a :

### WorkingPrecision     (2)

Compute the ArcLength using machine arithmetic:

In some cases, the exact answer cannot be computed:

Find the ArcLength using 30 digits of precision:

## Applications     (8)

The length of a function curve :

Compute the length of a knot:

Compute the length of Jupiter’s orbit in meters:

The length can be computed using the polar representation of an ellipse:

Alternatively, use elliptic coordinates with half focal distance and constant :

Extract lines from a graphic and compute their coordinate length:

Color a Lissajous curve by distance traversed:

Color Viviani’s curve on the sphere by the fraction of distance traversed:

Find mean linear charge density along a circular wire:

Compute the perimeter length of a Polygon :

## Properties & Relations     (6)

ArcLength is a non-negative quantity:

ArcLength [ r ] is the same as RegionMeasure [ r ] for any one-dimensional region:

ArcLength for a parametric form is defined as an integral:

ArcLength [ x , t , c ] is equivalent to RegionMeasure [ x , < t >, c ] :

For a 1D region, ArcLength is defined as the integral of 1 over that region:

The circumference of a 2D region is the ArcLength of its RegionBoundary :

## Possible Issues     (2)

The parametric form or ArcLength computes the length of possibly multiple coverings:

An arc of a circle is a "portion" of the circumference of the circle.

The length of an arc is simply the length of its "portion" of the circumference. The circumference itself can be considered a full circle arc length.

If we solve the proportion for arc length, and replace "arc measure"
with its equivalent "central angle", we can establish the formula:

Notice that arc length is a fractional part of the circumference. For example, an arc measure of 60º is one-sixth of the circle (360º), so the length of that arc will be one-sixth of the circumference of the circle.

In circle O, the radius is 8 inches and minor arc is intercepted by a central angle of 110 degrees. Find the length of minor arc to the nearest integer.

As you progress in your study of mathematics and angles, you will see more references made to the term "radians" instead of "degrees". So, what is a "radian" ?

which gives arc length, s : s = θr

subtend = "to be opposite to"

In a circle, the arc measure of the entire circle is 360º and the arc length of the entire circle is represented by the formula for circumference of the circle: .

Substituting C into the formula s = θr shows:
C = θr 2πr = θr 2π = θ
The arc measure of the central angle of an entire circle is 360º and the radian measure of the central angle of an entire circle = 2π.

3. Convert to degrees.

4. Convert to degrees.

5. Find the length of an arc subtended by an angle of radians in a circle of radius 20 centimeters.

The diagram at the right shows two circles with the same center (concentric circles). It has already been shown that concentric circles are similar under a dilation transformation.

The ratio of similitude of the smaller circle to the larger circle is:

As long as the central angles are the same, the slices (sectors) will be similar.

Since corresponding parts of similar figures are in proportion,
An equivalent proportion can be written as This proportion shows that the ratio of the arc length intercepted by a central angle to the radius of the circle will always yield the same (constant) ratio.

In relation to the two arc length formulas seen on this page, both show that arc length, s, is expressed as "some value" times the radius, r. The arc length is proportional to the radius.