WORKSHEET ON ARC LENGTH AND AREA OF SECTOR

Worksheet on Arc Length and Area of Sector :

Worksheet given in this section is much useful to the students who would like to practice problems on length of arc length and area of sector. 

Worksheet on Arc Length and Area of Sector - Questions

Question 1 :

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

Question 2 :

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)

Question 3 :

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)

Question 4 : 

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

Question 5 :

Find the area of the sector that is outlined with the bold line. (Take ∏  =  3.14 and round your answer to one decimal place, if necessary)

Question 6 :

In circle C, if XCZ is a central angle and XYZ is an inscribed angle and m∠XYZ  =  58° and radius is 10 inches. Find the area of sector XCZ. (Take ∏  =  3.14 and round your answer to one decimal place, if necessary)

Question 7 :

If QRS is a central angle and m∠QRS = 46°, m∠SRT = 80°, and diameter is 4 inches, then find the area of the shaded sector. (Take ∏  =  3.14 and round your answer to one decimal place, if necessary)

Arc Length Worksheet - Solution

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°) ⋅ 2Π 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.

Question 2 :

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°) ⋅ 2Π 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.

Question 3 :

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 ⋅ 19°

Arc Measure  =  38°

The formula to find the arc length is

=  (Arc Measure / 360°) ⋅ 2Π 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.

Question 4 : 

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°) ⋅ 2Π 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.

Question 5 :

Find the area of the sector that is outlined with the bold line. (Take ∏  =  3.14 and round your answer to one decimal place, if necessary)

Solution :

The formula to find area of the sector is

=  (θ / 360°) ⋅ Π r²

Plug r  =  11, θ  =  300° and Π  ≈  3.14

≈  (300° / 360°) ⋅ 3.14 ⋅ 11²

≈  316.7

Hence, the area of the given sector is about 316.7 cm².

Question 6 :

In circle C, if XCZ is a central angle and XYZ is an inscribed angle and m∠XYZ  =  58° and radius is 10 inches. Find the area of sector XCZ. (Take ∏  =  3.14 and round your answer to one decimal place, if necessary)

Solution :

By Inscribed Angle Theorem, we have

1/2 ⋅ m∠XCZ  =  m∠XYZ

Multiply both sides by 2.

m∠XCZ  =  2 ⋅ m∠XYZ

Given : m∠XYZ  =  58°.

Then, we have

m∠XCZ  =  2 ⋅ 58°

m∠XCZ  =  116°

So, the central angle θ is 116°.

The formula to find area of the sector is

=  (θ / 360°) ⋅ Π r²

Plug r  =  10, θ  =  116° and Π  ≈  3.14

≈  (116° / 360°) ⋅ 3.14 ⋅ 10²

≈  101.2

Hence, the area of sector XCZ is about 101.2 in².

Question 7 :

If QRS is a central angle and m∠QRS = 46°, m∠SRT = 80°, and diameter is 4 inches, then find the area of the shaded sector. (Take ∏  =  3.14 and round your answer to one decimal place, if necessary)

Solution :

Given : m∠QRS  =  46° and m∠SRT  =  80°.

Then, we have

m∠QRS + m∠SRT  =  46° + 80°

m∠QRS + m∠SRT  =  126°

Measure of central angle of the shaded region :

m∠QRT  =  360° - 126°

m∠QRT  =  234°

Radius of the circle :

Radius  =  Diameter / 2

Radius  =  4 / 2

Radius  =  2 inches

The formula to find area of the sector is

=  (θ / 360°) ⋅ Π r²

Plug r  =  2, θ  =  234° and Π  ≈  3.14

≈  (234° / 360°) ⋅ 3.14 ⋅ 2²

≈  8.2

Hence, the area of the shaded sector is about 8.2 in².

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

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