geometry problem: circle inscribed in a sector?

a circle having a diameter of 8 cm is inscribed in a sector of a circle whose angle is 80°. find the area of the sector.

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  • Brian
    Lv 7
    6 years ago
    Favorite Answer

    Since the two straight sides bordering the sector

    are tangent to the circle, we know that the right

    triangle formed by the centers of the two circles

    and the point of tangency of the smaller circle

    and either of the sector sides has the radius of

    the smaller circle as one leg and the angle opposite

    this leg being (1/2)*80 = 40 degrees. The hypotenuse

    of this triangle is then the distance between the two

    centers, and the radius of the larger circle will then be

    this hypotenuse length plus the radius of the smaller

    circle. So with the smaller circle having a radius of 4 cm,

    the hypotenuse has a length of 4*csc(40), and thus the

    radius of the larger circle is 4*csc(40) + 4 = 4*(1 + csc(40)) cm.

    The area of the sector is therefore

    (1/2)*r^2*(theta) = (1/2)*[4*(1 + csc(40))]^2 * (80 * (pi/180)) =

    8*(1 + csc(40))^2 * (4/9) * pi = (32/9)*pi*(1 + csc(40))^2 cm^2,

    which is 72.96 cm^2 to 2 decimal places.

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  • Hosam
    Lv 6
    6 years ago

    If you draw a sketch you'll find that the inscribed circle has its center on the bisector of the

    80 degree angle of the sector. If you draw perpendiculars from the center to the two lines defining

    the sector then you can express the distance between the center of the inscribed circle and the

    tip of the sector as

    r / x = tan(80/2) = tan (40)

    Therefroe, x = r / tan(40)

    The radius of the sector is x + r = r (1/tan(40) + 1) = 8 ( 1/ tan(40) + 1) = 17.534 cm

    Now we can find the area of the sector

    Area = 1/2 R^2 (theta), where theta is expressed in radians.

    Area = (1/2) (17.534)^2 ( 80 * pi /180 ) = 214.634 cm^2

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  • 6 years ago

    area of sector is given by 80 pie R^2 / 360 where R = 4(1 + csc 40) ANSWER

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  • 6 years ago

    Area of sector = (x/360)πr^2

    where x is central angle of sector

    => A = (80/360)π(8)^2

    = 44.68 sq.cm

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  • 6 years ago

    A = ( 1 / 2 )r^2 * theta

    Source(s): my brain
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