HOW TO FIND RADIUS WHEN ARC LENGTH IS GIVEN

How to find radius when arc length is given ?

Here we are going to see some examples problems to show, how to find radius when arc length is given.

Example 1 :

The arc length of a sector is 66 cm and the central angle is 30°. Find its radius.

Solution :

Arc length of a sector  =  66 cm

Central angle  =  30°

=  (θ/360) ⋅ 2Πr

66  =  (30/360) ⋅ 2 ⋅ (22/7) ⋅ r

(66 ⋅ 7 ⋅ 360) / (30 ⋅ 22 ⋅ 2)  =  r

r  =  126 cm

Hence the radius of the sector is 126 cm.

How to find radius, central angle and perimeter of sector ?

Example 2 :

Find the radius, central angle and perimeter of a sector whose arc length and area are 27.5 cm and 618.75 cm2 respectively.

Solution :

Given that l = 27.5 cm and Area  =  618.75 cm2. So,

Area  =  lr/2

=  618.75 cm2

(275 ⋅ r)/2  =  618.75

r  =  45 cm

Hence, perimeter is l + 2r = 27.5 + 2(45) = 117.5cm

Now, arc length is given by  (θ/360) ⋅ 2Πr  =  l

(θ/360) ⋅ 2   (22/7)  45  =  27.5

θ  =  35°

How to find radius when area and arc length is given ?

Example 3 :

Find the radius of the sector of area 225 cm2 and having an arc length of 15 cm

Solution :

Area of sector  =  225 cm ---(1)

Arc length  =  15 cm

Area of sector  =  lr/2  ---(2)

(1)  =  (2)

225  =  (15 ⋅ r)/2

(225  ⋅ 2)/15  =  r

r  =  30 cm

Hence the radius of sector is 30 cm.

Example 4 :

Find the radius, central angle and perimeter of a sector whose length of arc and area are 4.4 m and 9.24 m2 respectively

Solution :

Given that l  =  4.4 m and Area  =  9.24 m2. So,

Area  =  lr/2

=  9.24 m2

(4.4 ⋅ r)/2  =  9.24

r  =  (2 (9.24))/4.4

r  =  4.2

Hence, perimeter is l + 2r = 4.4 + 2(4.2) = 12.8 m

Now, arc length is given by  (θ/360) ⋅ 2Πr  =  l

(θ/360) ⋅ 2   (22/7)  4.2  =  4.4

θ  =  (4.4 ⋅ 7 ⋅ 360)/(2 ⋅ 22 ⋅ 4.2)

θ  =  60°

Hence the required angle and radius are 60° and 4.2 m respectively.

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