LENGTH OF ARC WORKSHEET

Question 1 :

Find the length of the arc whose radius is 42 cm and central angle is 60° (Take π  ≈  3.14 and round your answer to the nearest hundredth, if necessary).

Answer :

Arc length is

=  (θ/360°) ⋅ 2πr

Substitute r  =  42, θ  =  60° and π  ≈  3.14. 

≈  (60°/360°) ⋅ 2 ⋅ (3.14) ⋅ 42

=  (1/6) ⋅ 263.76

=  43.96

So, the length of the arc is about 43.96 cm. 

Question 2 :

Find the length of the arc whose radius is 10.5 cm and central angle is 36° (Take π  ≈  3.14 and round your answer to the nearest hundredth, if necessary).

Answer :

Arc length is

=  (θ/360°) ⋅ 2πr

Substitute r  =  10.5 and θ  =  36° and π  ≈  3.14.

≈  (36°/360°) ⋅ 2 ⋅ (3.14) ⋅ 10.5

 =  (1/10) ⋅ 65.94

=  6.59

So, the length of the arc is about 6.59 cm. 

Question 3 :

Find the length of the arc whose radius is 21 cm and central angle is 120° (Take π  ≈  3.14 and round your answer to the nearest hundredth, if necessary).

Answer :

Arc length is

=  (θ/360°) ⋅ 2πr

Substitute r  =  21 and θ  =  120° and π  ≈  3.14. 

≈  (120°/360°) ⋅ 2 ⋅ (3.14) ⋅ 21

 =  (1/3) ⋅ 131.8

=  43.96

So, the length of the arc is about 43.96 cm.

Question 4 :

Find the length of an arc, if the radius of circle is 14 cm and area of the sector is 63 square cm.

Answer :

Area of the sector  =  63 square cm

lr/2  =  63

Substitute r  =  14 cm. 

l(14)/2  =  63

l(7)  =  63

l  =  9 cm

So, the required arc length is 9 cm.

Question 5 :

Find the length of arc, if the perimeter of a sector is 45 cm and radius is 10 cm.

Answer :

Perimeter of sector  =  45 cm

l + 2r  =  45

Substitute r  =  10 cm.

l + 2(10)  =  45

l + 20  =  45

l  =  45 - 20

l  =  25 cm

Question 6 :

Find the arc length whose central angle is 180° and circumference of the circle is 64 cm.

Answer :

Circumference of circle  =  64 cm

2πr  =  64

Arc length is

l  =  (θ/360°) ⋅ 2πr

Substitute θ  =  180° and 2πr  =  64.

  l  =  (180°/360°) ⋅ 64

l  =  (1/2) ⋅ 64

l  =  32 cm

l  =  32 cm

Question 7 :

Find the area of the sector whose arc length is 20 cm and radius is 7 cm.

Answer :

Area of sector  =  lr/2

Substitute l  =  20 and r  =  7.

Area of sector  =  (20 x 7) / 2

Area of sector  =  70 square units.

Question 8 :

A pendulum swings through an angle of 30° and describes an arc length of 11 cm. Find the length of the pendulum.

Answer :

Arc length of sector  =  11 cm

sector angle  =  30°

If the pendulum swings once, then it forms a sector and the radius of the sector is the length of the pendulum.

So,

l  =  (θ/360°) x 2πr

Substitute the known values and solve for r.

11  =  (30°/360°) x 2 x (22/7) x r

 11  =  (1/12) x 2 x (22/7) x r

r  =  (11 x 7 x 12)/(2 x 22)

r  =  7 x 3

r  =  21 cm

So, the length of pendulum is 21 cm.

Question 9 :

The windshield wiper of a car is 24 inches long. How many inches will the tip of the wiper trace out in 1/4 revolution ?

Answer :

The length of the arc by 1/4 of the revolution.

= 1/4 of 2πr

r = 24

= 1/4 x 2π(24)

= 12π

Question 10 :

The exact circumference of a circle is 34π meters. What is the approximate area of the circle. Use π = 3.14

Answer :

Circumference of circle = 34π meter

2πr = 34π

2r = 34

r = 17 meter

Area of circle = πr²

= 3.14(17)²

= 3.14 x 289

= 907.46 square meter.

Question 11 :

Tye has a square piece of yellow felt that has an area of 81 square inches. She wants to cut the largest circle possible from the material to create a sum of her art project. What is the area of felt circle ?

Answer :

Area of square = 81 square inches

Diameter of the circle = side length of square

a² = 81

a = 9 inch

Diameter of the circle = 9

radius = 9/2

= 4.5 inches

Area of circle = πr²

= 3.14 (4.5)²

= 63.58 square inches

Question 12 :

Tarek has 72 feet of plastic fencing to make a flower garden in this backyard. The garden can either be circular or square. If he uses all of the fencing, what is the difference between area of the circular garden and square garden ?

Answer :

Length of fencing of square garden = 72 feet

4 x side length = 72

side length of square = 72/4

= 18 feet

Area of square field = 18 x 18

= 324 square feet

Circumference of circle = 2πr

2πr = 72

r = 72/2π

r = 36/π 

Area of circle = πr²

=  π(36/π)²

= 412.73

Difference = 412.73  - 324

= 88.73 square feet.

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