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Lesson Explainer: Arc Lengths Mathematics • 10th Grade

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

We recall that an arc of a circle is a portion of its circumference between two radii. If we know what proportion of the circumference the arc is, then we can use the length of the circumference of the circle to determine the length of the arc. For example, consider an arc of a quarter of a circle of radius 1.

First, recall that a circle of radius π‘Ÿ has circumference 2πœ‹π‘Ÿ. This means that the circumference of this circle is 2πœ‹(1)=2πœ‹.

Next, we know that the arc’s length is 14 of the circle’s circumference. So, length 𝑙 of the arc is 𝑙=14(2πœ‹)=πœ‹2.

We can see that this arc represents one-quarter of the circle, but it is good practice to see why this is the case. A full turn is an angle of 2πœ‹ rad, so a πœ‹2 rad angle is 2πœ‹=14οŽ„οŠ¨ of the circle. Therefore, the arc will subtend an angle of πœ‹2, and dividing this value by 2πœ‹ will tell us the proportion of the circle that is the arc.

Hence, the arc length is one-quarter of the circumference: 𝑙=Γ—=2πœ‹οŽΓ—(2πœ‹)=14(2πœ‹)=πœ‹2.proportioncircumferenceοŽ„οŠ¨

In general, if the arc subtends an angle of πœƒ rad, then the proportion of the circle that is the arc is πœƒ2πœ‹ and the circumference of the circle is 2πœ‹π‘Ÿ.

Hence, 𝑙=Γ—=ο€½πœƒ2πœ‹ο‰Γ—(2πœ‹π‘Ÿ)=ο€πœƒ2πœ‹οŒΓ—ο€Ό2πœ‹π‘Ÿοˆ=π‘Ÿπœƒ.proportioncircumference

We have shown the following result.

Property: Length of an Arc

Length 𝑙 of an arc of a circle of radius π‘Ÿ that subtends an angle πœƒ rad is given by 𝑙=π‘Ÿπœƒ.

In our first example, we will use the formula for the length of an arc to determine the length of an arc of a circular sector from its radius and central angle.

Example 1: Finding the Arc Length of a Segment

An arc has a measure of 2πœ‹3 radians and a radius of 9. Work out the length of the arc, giving your answer in terms of πœ‹ in its simplest form.

Answer

We first recall that if an arc has a measure of 2πœ‹3 radians, it is the same as saying that it subtends this angle. We can sketch the information given to better understand the question. We have a sector of a circle of radius 9 with a central angle of 2πœ‹3 rad.

We want to determine the length of the arc of this sector. We can do this by recalling that the length of the arc of a circle of radius π‘Ÿ that subtends an angle of πœƒ rad is given by π‘Ÿπœƒ.

Substituting π‘Ÿ=9 and πœƒ=2πœ‹3 into the formula gives us arclength=π‘Ÿπœƒ=9Γ—ο€Ό2πœ‹3.

Canceling the shared factor of 3 then gives 9Γ—ο€Ό2πœ‹3=3Γ—2πœ‹=6πœ‹.

We cannot simplify this any further. Hence, the length of the arc is 6πœ‹.

In our next example, we will use the length of an arc of a circle and the radius of the circle to determine the angle subtended by the arc in radians.

Example 2: Finding the Measure of the Angle of an Arc given Its Arc Length and Radius

An arc of a circle of radius 5 cm has a length of 3.5 cm. Find the angle subtended by the arc in radians.

Answer

We first recall that length 𝑙 of the arc of a circle of radius π‘Ÿ that subtends an angle of πœƒ rad is given by 𝑙=π‘Ÿπœƒ. We are told that π‘Ÿ=5 and that 𝑙=3.5. We can substitute these values into the equation to get 3.5=5πœƒ.

We can solve for πœƒ by dividing both sides of the equation by 5. We get πœƒ=3.55=0.7.rad

In our next example, we will find the radius of an arc using the length of the arc and the measure of the angle subtended by the arc.

Example 3: Finding the Radius of an Arc given Its Arc Length and the Measure of Its Angle

An arc of a circle has a length of 2.7 cm and it subtends and angle of 0.3 rad. Find the radius of the circle.

Answer

We first recall that length 𝑙 of the arc of a circle with radius π‘Ÿ that subtends an angle of πœƒ rad is given by 𝑙=π‘Ÿπœƒ. We are told that 𝑙=2.7cm and that πœƒ=0.3rad. We can substitute these values into the equation to get 2.7=π‘Ÿ0.3.

We can then divide both sides of the equation through by 0.3 to find π‘Ÿ. We get π‘Ÿ=2.70.3=9.cm

In our next example, we will use the formula for the arc length to determine the perimeter.

Example 4: Finding the Perimeter of a Shape Using Arc Lengths

Find the perimeter of 𝐴𝐡𝐢𝐷 in the following diagram.

Answer

Let’s start by highlighting 𝐴𝐡𝐢𝐷 on the diagram so that we can see the shape whose perimeter we are asked to calculate.

We can see that 𝐴𝐡𝐢𝐷 consists of two line segments and two arcs of a circle. We know that the perimeter of the shape will be equal to the sum of the lengths of the line segments 𝐴𝐡 and 𝐢𝐷 and arcs 𝐴𝐷 and 𝐡𝐢.

We are given that 𝐴𝐡=4cm, and we can find 𝐢𝐷 by noting that 𝑂𝐢 is a radius of a circle of length 7 cm and that 𝑂𝐷 is a radius of a circle of length 3 cm. Therefore, 𝐢𝐷=7βˆ’3=4.cm

We can determine the lengths of the arcs by recalling that length 𝑙 of the arc of a circle of radius π‘Ÿ that subtends an angle of πœƒ rad is given by 𝑙=π‘Ÿπœƒ. We can find the length of arc 𝐴𝐷 by noting sector 𝐴𝑂𝐷 has radius 3 cm and it subtends an angle of 1.6 rad.

Therefore, we can substitute π‘Ÿ=3cm and πœƒ=1.6rad into the arc length formula to get 𝐴𝐷=3Γ—1.6=4.8.cm

Similarly, we can note that arc 𝐡𝐢 is the arc of a circle of radius 7 cm that subtends an angle of measure 1.6 rad.

Thus, we can substitute π‘Ÿ=7cm and πœƒ=1.6rad into the arc length formula to get 𝐡𝐢=7Γ—1.6=11.2.cm

Finally, the sum of these four lengths gives us the perimeter of 𝐴𝐡𝐢𝐷.

We have 4+11.2+4+4.8=24.cm

Hence, 𝐴𝐡𝐢𝐷 has a perimeter of 24 cm.

In our final two examples, we will use arc lengths to solve real-world problems.

Example 5: Using Arc Length to Solve a Real-World Problem

The shape of the base of a bottle consists of a line segment of length 5 cm that is connected to the arc of a circle of radius 3 cm as shown.

Find the length of the arc of the circle to one decimal place.

Answer

To determine the length of the arc of the circle, we will use the formula 𝑙=π‘Ÿπœƒ, where 𝑙 is the length of the arc, π‘Ÿ is the radius of the circle, and πœƒ is the angle subtended by the arc in radians. We are given that the radius of the circle is 3 cm. However, we are not given the angle subtended by the arc.

We can determine one of the angles at the center of the circle by noting that we have a triangle with three known side lengths.

We can apply the cosine rule to determine the angle at 𝑂. The cosine rule tells us that π‘Ž=𝑏+π‘βˆ’2𝑏𝑐𝐴.cos

We want to determine the angle at 𝑂; we can call this 𝐴 in the diagram and label the side lengths π‘Ž, 𝑏, and 𝑐 as shown.

We then substitute π‘Ž=5cm, 𝑏=3cm, and 𝑐=3cm into the cosine rule to get 5=3+3βˆ’2(3)(3)𝐴.cos

Evaluating and simplifying, we have 25=9+9βˆ’18𝐴25=18βˆ’18𝐴.coscos

We can then subtract 18 from both sides of the equation to obtain 7=βˆ’18𝐴.cos

We then divide both sides of the equation by βˆ’18, giving us βˆ’718=𝐴.cos

We can find the value of 𝐴 by taking the inverse cosine of both sides of the equation, where it is important to check that our calculator is set to radians mode. This gives us 𝐴=ο€Όβˆ’718=1.97….cosrad

We can add this angle onto our original diagram.

There are now two different ways we can determine the arc length of this circle.

The first method we can use is to find the angle subtended by the arc by noting that the two angles at 𝑂 sum to give 2πœ‹.

Thus, cosοŠ±οŠ§ο€Όβˆ’718+πœƒ=2πœ‹.

We use the exact value of cosοŠ±οŠ§ο€Όβˆ’718to avoid rounding errors. We can rearrange and solve for πœƒ: πœƒ=2πœ‹βˆ’ο€Όβˆ’718=4.31….cosrad

We can now determine the arc length by substituting π‘Ÿ=3cm and πœƒ=4.31…rad into the arc length formula. We get 𝑙=3Γ—(4.31…)=12.93….cm

Once again, it is important to use the exact value for πœƒ to avoid rounding too early. We can now round this to one decimal place to reach our solution of 𝑙=12.9cm.

Let’s now look at the second method. In this method, we will find the length of the arc subtended by the angle of 1.97… rad.

We substitute π‘Ÿ=3cm and πœƒ=ο€Όβˆ’718cosrad into the arc length formula to get 𝑙=3ο€Όβˆ’718=5.91….coscm

We can find the circumference of the circle using the formula 𝑐=2πœ‹π‘Ÿ. Thus, 𝑐=2πœ‹(3)=6πœ‹cm

The length of the major arc in this circle is then the circumference minus the length of the minor arc. We find 6πœ‹βˆ’(5.91…)=12.93….cm

We use exact values in these calculations to avoid rounding errors.

Once again, we find that the length of the arc in the diagram is 12.9 cm to one decimal place.

Example 6: Using Arc Length to Solve a Real-World Problem

In soccer, opposing players must stand 10 yd away from the ball during a free kick and the goal is 8 yd wide. A free kick takes place such that the ball is 25 yd away from the goal with angles and points as those shown in the diagram. Point 𝐹 is where the ball lies, points 𝐢 and 𝐷 mark the edges of the goal, and points 𝐴 and 𝐡 mark the points on 𝐹𝐢 and 𝐹𝐷 that are 10 yd from 𝐹.

Given that 𝐹𝐴𝐡 is a sector, Find the perimeter of the shaded region to the nearest yard.

Answer

We can start by noting that the shaded region is bounded by three line segments and the arc of a circle, so the perimeter of the regions is the sum of the lengths of these line segments and the length of the arc.

We are given the lengths of two of the line segments in the question and diagram. We can find the length of the circular arc by recalling that length 𝑙 of the arc of a circle of radius π‘Ÿ that subtends an angle of πœƒ rad is given by 𝑙=π‘Ÿπœƒ. We see that the radius of the circle is 10 yd and the arc subtends an angle of 0.3 rad. Substituting these values into the arc length formula yields 𝑙=10(0.3)=3.yd

To determine the length of the final side of the region, we can note that it is part of a triangle with two known side lengths and a known angle.

We can find the missing side length of the triangle using the cosine rule: 𝑐=π‘Ž+π‘βˆ’2π‘Žπ‘πΆ.cos

We have 𝐢=1.67rad, π‘Ž=25yd, and 𝑏=8yd. Substituting these values into the cosine rule gives us 𝑐=25+8βˆ’2(25)(8)(1.67).cos

Evaluating gives us 𝑐=625+64βˆ’400(1.67).cos

We can now take the square roots of both sides of the equation, noting that π‘Ž must be positive, to get 𝑐=√625+64βˆ’400(1.67)β‰ˆ26.99….cos

This is the length of the entire side of the triangle. Subtracting 10 yd from this gives 16.99… yd. We can now add the lengths to the diagram.

Adding all of these lengths gives us the perimeter of the region: perimeteryd=3+15+8+16.99β€¦β‰ˆ43.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • An arc of a circle of radius π‘Ÿ that subtends an angle πœƒ measured in radians has a length of π‘Ÿπœƒ.
  • Perimeter 𝑃 of a circular sector that subtends an angle πœƒmeasured in radians is 𝑃=2π‘Ÿ+π‘Ÿπœƒ.

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