**Arc Length of a Sector :**

In this section, we are going to see how to calculate the length of an arc of a circle or part of the circle's circumference.

The formula given below can be used to calculate the length of an arc of a circle.

**Question 1 :**

Find the length of the arc that is bolded. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)

**Solution :**

**The formula to find the arc length is**

**= (Arc Measure / 360**°**) **⋅ 2**Π r**

**Plug r = 8, Arc Measure**** = 135**° and **Π **≈** 3.14**

≈ (135° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 8

≈ 18.9

Hence, the length of the arc is about 18.9 yd.

Let us look at the next problem on "Arc length of a sector".

**Question 2 :**

Find the length of the arc that is bolded. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)

**Solution :**

**The formula to find the arc length is**

**= (Arc Measure / 360**°**) **⋅ 2**Π r**

**Plug r = 8, Arc Measure**** = 315**° and **Π **≈** 3.14**

≈ (315° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 8

≈ 44

Hence, the length of the arc is about 44 cm.

Let us look at the next problem on "Arc length of a sector".

**Question 3 :**

Find the length of the arc highlighted in red color. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)

**Solution :**

**Given : **Diameter is 4 inches.

**Then, the radius is **

**= Diameter / 2**

**= 4 / 2 **

**= 2 inches**

**The formula to find the arc length is**

**= (Arc Measure / 360**°**) **⋅ 2**Π r**

**Plug r = 2, Arc Measure**** = 80**° and **Π **≈** 3.14**

≈ (80° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 2

≈ 2.8

Hence, the length of the arc is about 2.8 inches.

Let us look at the next problem on "Arc length of a sector".

**Question 4 :**

In the diagram given below, if QRS is a central angle and m∠QRS = 81°, m∠SRT = 115°, and radius is 5 cm, then find the length of the arc QST. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)

**Solution :**

**To find the length of the arc QST, first we have to find the arc measure QST or the central angle **m∠QRT.

m∠QRT = m∠QRS + m∠SRT

m∠QRT = **81**° + 115°

m∠QRT = 196°

**The formula to find the arc length is**

**= (Central Angle / 360**°**) **⋅ 2**Π r**

**Plug r = 5, Central Angle**** = 196**° and **Π **≈** 3.14**

≈ (196° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5

≈ 17.1

Hence, the length of the arc is about 17.1 cm.

Let us look at the next problem on "Arc length of a sector".

**Question 5 :**

If m∠LMN = 19° and radius is 15 inches, then find the length of arc LN. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)

**Solution :**

**To find the length of the arc LN, first we have to find the arc measure LN**.

**By Inscribed Angle Theorem, we have**

**1/2 **⋅ Arc Measure = **m∠LMN**

**Multiply both sides by 2. **

**Arc Measure = ****2 **⋅ **m∠LMN**

**Arc Measure = ****2 **⋅ 1**9**°

**Arc Measure = 38****°**

**The formula to find the arc length is**

**= (Arc Measure / 360**°**) **⋅ 2**Π r**

**Plug r = 15, Arc Measure**** = 38**° and **Π **≈** 3.14**

≈ (38° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 15

≈ 9.9

Hence, the length of the arc is about 9.9 inches.

Let us look at the next problem on "Arc length of a sector".

**Question 6 :**

In a circle, if the arc length of Arc AB is 18 cm and the measure of Arc AB is 39°, then find the radius of the circle. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)** **

**Solution :**

**Given : **The arc length of Arc AB is 18 cm.

So, we have

**(Arc Measure / 360**°**) **⋅ 2**Π r = 18**

**Plug Arc Measure**** = 39**° and **Π **≈** 3.14**

(39° / 360°) ⋅ 2 ⋅ 3.14 ⋅ r ≈ 18

(39° / 360°) ⋅ 2 ⋅ 3.14 ⋅ r ≈ 18

0.1083 ⋅ 2 ⋅ 3.14 ⋅ r ≈ 18

0.68 ⋅ r ≈ 18

Divide both sides by 0.68.

r ≈ 18 / 0.68

r ≈ 26.5

Hence, the radius of the circle is about 26.5 cm.

Let us look at the next problem on "Arc length of a sector".

**Question 7 :**

In a circle, if the arc length of Arc AB is 19 inches and the radius is 29 inches, then find the measure of arc AB. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)** **

**Solution :**

**Given : **The arc length of Arc AB is 19 inches.

So, we have

**(Arc Measure / 360**°**) **⋅ 2**Π r = 19**

**Plug r = 29 **and **Π **≈** 3.14**

(Arc Measure / 360°) ⋅ 2 ⋅ 3.14 ⋅ 29 ≈ 19

(Arc Measure / 360°) ⋅ 182.12 ≈ 19

(Arc Measure / 360°) ⋅ 182.12 ≈ 19

Multiply both sides by 360° / 182.12

Arc Measure ≈ 19 ⋅ 360° / 182.12

Arc Measure ≈ 19 ⋅ 360° / 182.12

Arc Measure ≈ 37.6°

Hence, the measure of arc AB is about 37.6°.

Let us look at the next problem on "Arc length of a sector".

**Question 8 :**

Find the length of the arc highlighted in red color. (Take ∏ ≈ 3.14 and round your answer to one decimal place, if necessary)

**Solution :**

**From the given diagram, we have**

**m∠MCN + Measure of arc MON = 360**°

**Plug ****m∠MCN = 88**°

**88**° + **Measure of arc MON = 360**°

Subtract 88° from both sides.

**Measure of arc MON = 272**°

**Given : **Diameter is 4 inches.

**Then, the radius is**

**= Diameter / 2**

**= 10 / 2**

**= 5 ft**

**The formula to find the arc length is**

**= (Arc Measure / 360**°**) **⋅ 2**Π r**

**Plug r = 5, Arc Measure**** = 272**° and **Π **≈** 3.14**

≈ (272° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5

≈ 23.7 ft

Hence, the length of the arc is about 23.7 ft.

After having gone through the stuff given above, we hope that the students would have understood how to find arc length.

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