How to Find the arc length when thetha is equal to 135 and sector area is 162cm squared?

Method 1 :

r is the radius of the sector.
Take π = 3.14

Sector area = 162 cm²
π × r² × (135/360) = 162
r² = 162 × 360 / 135π cm²
r = √(162 × 360 / 135π) cm

Arc length
= 2 × π × r × (135/360) cm
= 2 × π × √(162 × 360 / 135π) × (135/360) cm
= 2 × √(162 × 360π / 135) × (135/360) cm
= 2 × √(162 × 360 × 3.14 / 135) × (135/360) cm
= 27.6 cm (to 3 sig. fig.)


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Method 2 :

r, L and A are the radius, arc length and area of the sector respectively.
Take π = 3.14

L = 2 × π × r × (135/360)
L² = 4 × 3.14² × r² × (135/360)² …… [1]

A = π × r² × (135/360)
A = 3.14 × r² × (135/360) …… [2]

[1]/[2] :
L²/A = 4 × 3.14 × (135/360)
L² = A × 4 × 3.14 × (135/360)
L = √[A × 4 × 3.14 × (135/360)]
L = √[162 × 4 × 3.14 × (135/360)] cm
Arc length, L = 27.6 cm (to 3 sig. fig.)

Convert θ to radians.
θ = 135°×π/180° = 0.75π radians

r = radius of circle
area of circle = πr²
area of sector = πr²×θ/(2π) = πr²0.375
πr²0.375 = 162 cm²
r² = 162/(0.375π) = 432/π
r = √(432/π) = 12√(3/π)

arc length = rθ = 12√(3/π)0.75π = 9√(3π) ≅ 27.63 cm

Let S be the arc length; A the angle at the center; r the radius of the circle containing the arc.
S=rA-------(1)
Area=(r^2)A/2--------(2)
Putting (1) into (2) & eliminating r, get
Area=[(S/A)^2]A/2
=>
Area=(S^2)/(2A)
Now, A=135*, area=162 cm^2
=>
S=sqr[2(135*pi/180)*162)]
=>
S=27.63 cm approximately

First finf 'r' (radius)
A(sect) = (x/360)pi r^2
162 = (135/360) pi r^2
r^2 = (162 x 360) / (135 x 3.141592....)
r^2 = 137.509...
r = sqrt(137.509...) = 11.726...

Then
C = 2pi r
C(arc) = (135/360) x 2 x pi X 11.726... ....
C(Arc) = 27.6298... cm

　
A = area = 162 cm²
s = arc length
r = radius
θ = angle (in radians) = 3π/4

s = rθ

A = 1/2 r²θ
A = 1/2 (rθ)²/θ
A = 1/2 s²/θ
s² = 2Aθ
s² = 2 * 162 cm² * 3π/4 = 243π cm²
s = √(243π cm²) = 9√(3π) cm ≈ 27.63 cm

The area of the entire circle is A = pi*r^2
If the central angle is 135 degrees, the area of the sector is:
(135/360)*pi*r^2 = 162
0.375*pi*r^2 = 162
r^2 = 162/(0.375*pi)
r^2 = 432/pi
r = sqrt(432/pi), note that the radius r can't be negative
The circumference of the entire circle is:
C = 2*pi*r = 2*pi*sqrt(432/pi) = sqrt(1728*pi)
The length of the arc is:
(135/360)*sqrt(1728*pi)
= sqrt(243*pi)
=~ 27.62982111455518918894789387201