LENGTH OF ARC

Arc is a portion of edge of the circle defined by two end points.

There are two types of arcs. They are minor arc and major arc. 

Minor arc is an arc which has the central angle measure less than 180°. 

Major arc is an arc which has the central angle measure greater than 180°. 

Arc length of a sector is the length of the portion on the circumference of the circle intercepted between the bounding radii and is denoted by l.

Formula to find the length of the arc is 

 l  =  (θ/360°) x 2πr

where

l  = Length of the arc

θ  =  Arc measure or central angle

Formula for area of sector is

=  lr/2 square units

Formula for perimeter of sector is

=  l + 2r units

Examples

Example 1 :

Find the length of arc whose radius is 7 cm and central angle 90 degree.

Solution :

Arc length is

=  (θ/360°) ⋅ 2πr

Substitute r  =  42 and θ  =  60°. 

=  (60°/360°) ⋅ 2 ⋅ (22/7) ⋅ 7

 =  (1/4) ⋅ 2 ⋅ 22

=   (1/2) ⋅ 22

 =   11

So, the length of the arc is 11 cm.

Example 2:

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

Solution :

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.

Example 3 :

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

Solution :

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

Example 4 :

Find the arc length whose central angle is 180 degree and perimeter of circle is 64 cm.

Solution :

Perimeter of circle  =  64 cm

That is, 

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

Example 5 :

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

Solution :

Area of sector  =  lr/2

Substitute l  =  20 and r  =  7.

Area of sector  =  (20 x 7) / 2

Area of sector  =  70 square units.

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