WORKSHEET ON ARC LENGTH AND AREA OF SECTOR

Question 1 :

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

Detailed Answer Key

Question 1 :

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

Solution :

The formula to find the arc length is

=  (Arc Measure / 360°) ⋅ 2πr

Substitute r  =  8, arc Measure  =  315° and π  ≈  3.14.

≈  (315° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 8

=  44

So, 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 π ≈ 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°) ⋅ 2πr

Substitute r  =  5, Central Angle  =  196° and π  ≈  3.14.

≈  (196° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5

=  17.1

So, 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 π ≈ 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°) ⋅ 2πr

Substitute r  =  15, Arc Measure  =  38° and π  ≈  3.14.

≈  (38° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 15

=  9.9

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

Question 4 : 

Find the length of the arc highlighted in red color. (Take π ≈ 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°) ⋅ 2πr

Substitute r  =  5, Arc Measure  =  272° and Π  ≈  3.14.

≈  (272° / 360°) ⋅ 2 ⋅ 3.14 ⋅ 5

=  23.7 ft

So, 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 π ≈ 3.14 and round your answer to one decimal place, if necessary)

Solution :

The formula to find area of the sector is

=  (θ / 360°) ⋅ πr²

Substitute r  =  11, θ  =  300° and π  ≈  3.14.

≈  (300° / 360°) ⋅ 3.14 ⋅ 11²

=  316.7

So, 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 π ≈ 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°) ⋅ πr²

Substitute r  =  10, θ  =  116° and Π  ≈  3.14.

≈  (116° / 360°) ⋅ 3.14 ⋅ 10²

=  101.2

So, 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 π ≈ 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°) ⋅ πr²

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

≈  (234° / 360°) ⋅ 3.14 ⋅ 2²

=  8.2

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

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