a central angle in a circle has a measure of 34.4 and the measure of the area of its sector is 139.6 square units. what is the radius?

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a central angle in a circle has a measure of 34.4 and the measure of the area of its sector is 139.6 square units. to the nearest tenth of a unit, what is the measure of the circle's radius?a. 19.4b. 21.6c. 27.3d. 29.7

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  • 1 month ago
    Favorite Answer

    The area of a full circle is:

    A = πr²

    Given a full circle is 360°, the area of a sector (with central angle θ) is a fraction of full circle --> θ/360°

    So the area of a sector is:

    A(sector) = θ/360° * πr²

    Plug in your values:

    139.6 = 34.4/360 * πr²

    r² = 139.6 * 360/(34.4π)

    r² ≈ 465.03

    r ≈ √465.03

    r ≈ 21.6 units

  • TomV
    Lv 7
    1 month ago

    34.4 without dimensions should be interpreted as radians except I'm fairly sure you meant 34.4°.

    Working in radians, the relationship between central angle, Θ, radius, r, and circular sector area, A, can be stated as:A = Θr²/2

    Converting the angle to degrees, use the conversion:

    Θ (degrees) = Θ (radians)(180/π)

    or

    Θ (radians) = (π/180)Θ (degrees)

    Using this conversion, and working in degrees:

    A = (π/180)Θr²/2 = (Θ/360)πr²

    In either case, you should be able to recognize that the sector area is the same proportion of the area of a full circle as the sector angle is to the angular dimension of a full circle. 

    As/Ac = Θ/(2π) = Θ°/360°

    Plugging in the given values:

    A = (34.4/360)πr² = 139.6

    r = √[139.6*360/(34.4π)]

    r = 21.6 (rounded to 3 sig. dig)

  • Ian H
    Lv 7
    1 month ago

    Sector area A = (θ/360)πr^2

    r^2 = (360/θ)(A/π) = (360/34.4)(139.6/π) ~ 465.02

    r ~ 21.56 units, (nearest to b)

  • ?
    Lv 6
    1 month ago

    Area of sector working in degrees

    A = (x/360) pi r^2

    x = central angle

    Plug in A and x and solve for r

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