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

So, the radius of the sector is 126 cm.

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°

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 cm²  ---(1)

Arc length  =  15 cm

Area of sector  =  lr/2  ---(2)

(1)  =  (2)

225  =  (15 ⋅ r)/2

(225  ⋅ 2)/15  =  r

r  =  30 cm

So, 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°

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

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