In a circle a sector has an area of 16 cm2 and an arc length of 6.0 cm. what is the measure of the central angle in degrees?

Question

S=r(thera) and Area of sector is equal to 1/2radiussquaredtheta i dont know how to solve this because only the radius and the area are given

Answer 1

Hi Pia,

 

Let's begin by writing the formulas for sector area and arc length in terms of the central angle (theta) and the radius (r):

    sector area = (1/2)θr²

    arc length = θr

 

Next, let's find equivalent expressions for the central angle (theta) in terms of the radius (r) by manipulating the above formulas and setting these expressions equal to each other and then solving for the radius (r):

    sector area = (1/2)θr²  [sector area formula]

    16 = (1/2)θr²              [sector area plug-in]

    θr² = 32

    θ = 32/r²

    arc length = θr            [arc length formula]

    6 = θr                        [arc length plug-in]

    θ = 6/r

    32/r² = 6/r

    32/r = 6

    r = 32/6

    r = 16/3 cm

 

Next, we can plug in our value for the radius (r) in either formula and then solve for the central angle (theta) in radians (we'll use the easier math formula for arc length):

    arc length = θr  [arc length formula]
    6 = θ(16/3)       [arc length plug-ins]

    θ = 6/(16/3)

    θ = 6(3/16)

    θ = 18/16

    θ = 9/8 radians 

    

Finally, we can convert our answer in radians to degrees:

    (9/8 radians)(180º / pi radians)

    (9/8)(180) = 1620/8 = 405/2 = 202.5º

 

The key here was getting equivalent expressions for the central angle (theta) in terms of the radius (r) by manipulating our formulas, so that we could first determine the radius (r) and then determine the central angle (theta).

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.