Lesson Explainer: Arc Lengths Mathematics

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 180∘, or πœ‹ 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 180∘ or πœ‹ radians, or equivalent, when the radii form a diameter.

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.

The circumference, 𝐢, of a circle of radius π‘Ÿ is given by 𝐢=2πœ‹π‘Ÿ.

The length of the minor arc above can be calculated by multiplying the circumference, 2πœ‹π‘Ÿ, by 14. In general, for an arc with central angle πœƒ, this is a πœƒ360 section of the circumference and its length is calculated as arclength=2πœ‹π‘ŸΓ—πœƒ360=2πœ‹π‘Ÿπœƒ360.

We can do the same for an angle measured in radians. If the central angle is πœƒ radians, then the arc is a πœƒ2πœ‹ section of the circumference. Thus, the length of the arc is given by arclength=2πœ‹π‘Ÿπœƒ2πœ‹=π‘Ÿπœƒ.

This gives us the following formulas for finding the length of an arc of a circle.

Definition: Arc Length

The length of an arc that subtends an angle πœƒ, measured in degrees, in a circle of radius π‘Ÿ is given by arclength=2πœ‹π‘Ÿπœƒ360.

The length of an arc that subtends an angle πœƒ, measured in radians, in a circle of radius π‘Ÿ is given by arclength=π‘Ÿπœƒ.

We will now see some examples of how we can apply these formulas, beginning with how we can find an arc length given an angle in radians.

Example 1: Calculating the Length of an Arc

Find the length of the blue arc given the radius of the circle is 8 cm, and the angle measure shown is in radians. Give the answer to one decimal place.

Answer

In this problem, we are given the angle subtended by an arc whose measure is in radians. We recall that the length of an arc that subtends an angle πœƒ, measured in radians, in a circle of radius π‘Ÿ is given by arclength=π‘Ÿπœƒ.

We are given that the radius of this circle is 8 cm. Thus, we can substitute π‘Ÿ=8 and πœƒ=4πœ‹3 into the formula to give arclengthcm=8Γ—4πœ‹3=32πœ‹3.

As we are asked for an answer to the nearest decimal, we can use a calculator to find a decimal equivalent and approximate, which gives arclengthcmcm=33.510β€¦β‰ˆ33.5.

Therefore, to one decimal place, the length of the blue arc is 33.5 cm.

In the next example, we will find the length of an arc in a real-world context.

Example 2: Solving an Applied Problem Involving the Arc Length of a Pendulum

A pendulum of length 26 cm swings 58∘. Find the length of the circular pathway that the pendulum makes giving the answer in centimetres in terms of πœ‹.

Answer

In this question, we are given that the pendulum follows a circular path. This means that we can model the path of the pendulum as an arc of a circle. The pendulum pivots about a single point; thus, the length of the pendulum will be the radius of the circle. We are given the central angle of the arc as 58∘.

The arc length of a circle of radius π‘Ÿ with a central angle πœƒ, measured in degrees, is given by arclength=2πœ‹π‘Ÿπœƒ360.

So, substituting the values π‘Ÿ=26 and πœƒ=58∘ and simplifying, we have arclengthcm=2πœ‹(26)(58)360=3016πœ‹360=377πœ‹45.

We can leave our answer in terms of πœ‹ to give the length of the circular pathway as 37745πœ‹ cm.

As an alternative method, we can convert the angle in degrees to one in radians and then use the formula to find the length of an arc subtending an angle in radians. We recall that to change any angle in degrees to one in radians, we multiply the angle by πœ‹180. Hence, 58=58πœ‹180.∘

The length of an arc that subtends an angle πœƒ, measured in radians, in a circle of radius π‘Ÿ is given by arclength=π‘Ÿπœƒ.

Since the pendulum forms a circle of radius π‘Ÿ=26cm, we can substitute this into the formula, giving arclengthcm=26Γ—58πœ‹180=1508πœ‹180=37745πœ‹.

Either method will allow us to calculate the length of the circular pathway as 37745πœ‹ cm.

We can expand the process of finding the length of an arc of a circle to finding the perimeter of a circular sector. A sector of a circle is a part of the circle enclosed by two radii and an arc between them. We can recall that the perimeter of a shape is the distance around the outside edge.

The perimeter of a sector is the sum of the two radii and the arc length. We can define this below.

Definition: Perimeter of a Sector

The perimeter of the sector of a circle of radius π‘Ÿ subtended by an angle of πœƒ measured in degrees is perimeter=2πœ‹π‘Ÿπœƒ360+2π‘Ÿ.

The perimeter of the sector of a circle of radius π‘Ÿ subtended by an angle of πœƒ measured in radians is perimeter=π‘Ÿπœƒ+2π‘Ÿ.

In the next example, we will see how we can find the perimeter of a circular sector by first finding the length of the arc.

Example 3: Finding the Perimeter of a Sector

The radius of a circle is 7 cm and the central angle of a sector is 40∘. Find the perimeter of the sector to the nearest centimetre.

Answer

We can sketch this circular sector in the following way.

The perimeter of the sector, the distance around the outside edge, is the sum of the lengths of the two radii along with the arc that makes the sector: perimeterarclength=2π‘Ÿ+.

We are given that the length of the radius is 7 cm, but we will need to calculate the length of the arc.

The arc length of a circle of radius π‘Ÿ with a central angle πœƒ, measured in degrees, is given by arclength=2πœ‹π‘Ÿπœƒ360.

We are given that π‘Ÿ=7 and the central angle πœƒ=40∘. Hence, substituting these into the formula above gives arclengthcm=2πœ‹(7)(40)360=560πœ‹360=14πœ‹9.

We can keep this value in terms of πœ‹ for the next part of the calculation.

To find the perimeter, we substitute the radius π‘Ÿ=7 and arc length =14πœ‹9 into the perimeter calculation: perimeterarclength=2π‘Ÿ+.

Thus, we have perimetercm=2(7)+14πœ‹9=14+14πœ‹9=18.886….

We can then approximate this value to the nearest centimetre to give the result that the perimeter of the sector is 19 cm.

In the next example, we will use information about the perimeter of a sector to find its radius.

Example 4: Finding the Radius of a Circular Sector When Given the Central Angle and the Perimeter of the Sector

The perimeter of a circular sector is 67 cm and the central angle is 0.31 rad. Find the radius of the sector giving the answer to the nearest centimetre.

Answer

The perimeter of a sector is the distance around its outside edge. It is the length of two radii along with the arc that makes the sector. We can define the arc length as 𝑙 and write that perimeter=2π‘Ÿ+𝑙.

We are given that the perimeter is 67 cm, so we have the equation 67=2π‘Ÿ+𝑙.

We can use the information about the central angle of the sector to help us calculate the arc length, 𝑙, noting that the angle measure is in radians. We recall that the length of an arc, 𝑙, that subtends an angle πœƒ, measured in radians, in a circle of radius π‘Ÿ is given by 𝑙=π‘Ÿπœƒ.

We now substitute the given angle, πœƒ=0.31, into this equation to find 𝑙 as 𝑙=π‘ŸΓ—0.31=0.31π‘Ÿ.

Next, substituting 𝑙=0.31π‘Ÿ into the equation 2π‘Ÿ+𝑙=67 gives 2π‘Ÿ+0.31π‘Ÿ=672.31π‘Ÿ=67π‘Ÿ=672.31=29.004….cm

Finally, rounding to the nearest centimetre, we can give the answer that the radius of the sector is 29 cm.

In the final example, we will see how we can use information about tangents intersecting to find the length of an arc.

Example 5: Finding the Length of an Arc given Two Intersecting Tangents and Their Angle of Intersection

If π‘šβˆ π΄=76∘ and the radius of the circle equals 3 cm, find the length of the major arc 𝐡𝐢.

Answer

The major arc 𝐡𝐢 will be the larger of the two arc lengths, as shown in the following diagram.

In order to find the length of either the major or the minor arc 𝐡𝐢, we will need to establish the measure of the central angle subtending the arc. We can sketch the radii from 𝐡 and 𝐢 to the center, 𝑂, of the circle.

We recall that a tangent to the circle at a point 𝑃 meets the radius of the circle from 𝑃 at 90∘, so π‘šβˆ π΅=90∘ and π‘šβˆ πΆ=90∘. We can add this information to the diagram, along with the given information that π‘šβˆ π΄=76∘.

We observe that we now have a quadrilateral, 𝐴𝐡𝑂𝐢, and three of the angle measures within it. The sum of the internal angles in a quadrilateral is 360∘; thus, π‘šβˆ π΄+π‘šβˆ π΅+π‘šβˆ π‘‚+π‘šβˆ πΆ=360.∘

Substituting in the values for the angles and simplifying, we have 76+90+π‘šβˆ π‘‚+90=360256+π‘šβˆ π‘‚=360π‘šβˆ π‘‚=360βˆ’256=104.∘∘∘∘∘∘∘∘∘

We can now use this information, that π‘šβˆ π‘‚=104∘, to help us find the length of the major arc 𝐡𝐢.

The arc length of a circle of radius π‘Ÿ with a central angle πœƒ, measured in degrees, is given by arclength=2πœ‹π‘Ÿπœƒ360.

Here, if we use πœƒ=104∘, then this will give us the length of the minor arc 𝐡𝐢. There are two alternative options to find the length of the major arc. In the first method, we find the reflex βˆ π‘‚ by calculating 360βˆ’104=256∘∘∘. Substituting πœƒ=256∘ and π‘Ÿ=3 into the formula gives majorarclengthcm=2πœ‹(3)(256)360=1536πœ‹360=64πœ‹15.

We can keep this value in terms of πœ‹, or alternatively we can find the decimal equivalent as 13.404… cm and round to one decimal place to give the answer for the length of the major arc 𝐡𝐢 as 13.4 cm.

In the second, alternative method, when we initially calculated that π‘šβˆ π‘‚=104∘, we could find the length of the minor arc 𝐡𝐢 by substituting πœƒ=104∘, along with the radius, π‘Ÿ=3, to give the length of the minor arc as minorarclengthcm=2πœ‹(3)(104)360=26πœ‹15.

If we have the length of either arc and we wish to find the other, we can recognize that the sum of the two arcs is equal to the circumference of the circle. For a circle of radius π‘Ÿ, the circumference, 𝐢, is given by 𝐢=2πœ‹π‘Ÿ.

To find the circumference, we substitute the radius π‘Ÿ=3 into the formula, giving 𝐢=2πœ‹Γ—3=6πœ‹.cm

Now, to find the length of the major arc 𝐡𝐢, we can calculate majorarclengthcircumferenceminorarclengthcm=βˆ’=6πœ‹βˆ’26πœ‹15=64πœ‹15.

Either method has demonstrated that the length of the major arc 𝐡𝐢, rounded to one decimal place, is 13.4 cm.

We now summarize the key points.

Key Points

  • An arc of a circle is a section of the circumference of the circle between two radii.
  • The larger of the two arcs is the major arc and the smaller is the minor arc. If the central angle between the two radii is less than 180∘, or πœ‹ radians, then it is a minor arc. If it is larger than these values, then it is a major arc. If the angle is exactly 180∘, or πœ‹ radians, then there will be two semicircular arcs.
  • The length of an arc that subtends an angle πœƒ, measured in degrees, in a circle of radius π‘Ÿ is given by arclength=2πœ‹π‘Ÿπœƒ360.
  • The length of an arc that subtends an angle πœƒ, measured in radians, in a circle of radius π‘Ÿ is given by arclength=π‘Ÿπœƒ.
  • The perimeter of a sector is the sum of the lengths of two radii along with the arc that makes the sector.

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