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Area Sector

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heshamelaziry GMAT Destroyer! Default Avatar
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Area Sector Post Sat Nov 28, 2009 9:23 pm
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  • Lap #[LAPCOUNT] ([LAPTIME])
    The area of a certain circle is 8 times the area of a sector of the circle. What is the perimeter of the sector ?

    A- The circumference of the circle is 12Pi.

    B- The area of the sector is 9Pi/2

    OA D

    How to solve this considering the 2 equations:

    Area of a sector = (central angle/360) *PiR^2

    Length of sector arc = (central angle/ 360) * 2Pi r


    Please be detailed.

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    Stuart Kovinsky GMAT Instructor
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    Post Sat Nov 28, 2009 10:14 pm
    heshamelaziry wrote:
    The area of a certain circle is 8 times the area of a sector of the circle. What is the perimeter of the sector ?

    A- The circumference of the circle is 12Pi.

    B- The area of the sector is 9Pi/2

    We know that everything about sectors of circles is proportional. We have the general circle sector ratio:

    (degree measure of sector)/360 = (arc length of sector)/circumference = (area of sector)/(area of circle)

    From the original, we know that (area of sector)/(area of circle) = 1/8; therefore, we can figure out the degrees of the arc (360/8) and the relationship between the arc length and the circumference (1:8).

    What we're missing right now is a measurement - our circle could be tiny or could be huge.

    The GMAT loves to test circles in data sufficiency, because if you know 1 concrete measurement you automatically know all the others; if you know 1 of radius, diameter, circumference or area, you know the entire circle.

    So, we need pretty much any concrete measurement, keeping in mind that the perimeter of the sector is r + r + arc length of the sector.

    (1) circumference = 12pi. Perfect, we can calculate radius and the arc length of the sector: sufficient.

    (2) area of the sector = 9pi/2. Perfect, we can calculate the area of the entire circle, then the radius, then the circumference, then the arc length of the sector: sufficient.

    Each statement is sufficient alone: choose D.

    * * *

    Based on your question, it looks like you want to see the calculations as well. We'd never need (or want) to do so on a DS question, but here you go:

    (1) 2(pi)r = 12pi
    r = 6
    12pi = 8(arc length)
    1.5pi = arc length

    perimeter of sector = 2r + arc length = 12 + 1.5pi

    (2) area of the sector = 9pi/2
    area of circle = 8(9pi/2) = 36pi
    36pi = pi(r^2)
    36 = r^2
    6 = r
    circumference of circle = 2pi(r) = 12pi
    12pi = 8(arc length)
    1.5pi = arc length

    perimeter of sector = 2r + arc length = 12 + 1.5pi

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    heshamelaziry GMAT Destroyer! Default Avatar
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    Post Sat Nov 28, 2009 10:34 pm
    why 12Pi = 8 (arc length) ? i understood from your post that sector areas and arc lenghts are proportional to the circle area, but I do not understand the concept. Could you please elaborate a bit more on this concept relevant to 12Pi = 8(arc length) ?


    Thank you

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    Post Sat Nov 28, 2009 11:48 pm
    heshamelaziry wrote:
    why 12Pi = 8 (arc length) ? i understood from your post that sector areas and arc lenghts are proportional to the circle area, but I do not understand the concept. Could you please elaborate a bit more on this concept relevant to 12Pi = 8(arc length) ?


    Thank you
    It goes back to the original ratio:

    (sector area)/(area of circle) = (arc length)/circumference

    This ratio acknowledges that when we create a sector of the circle, we're simply dividing the area and circumference by the same amount.

    For example, if we create 4 sectors, each with a central angle of 90 degrees, each sector will contain 1/4 the area of the circle and the arc length along the edge of each sector will be 1/4 of the circumference.

    We know that for this circle:

    (sector area)/(area of circle) = 1/8, so we simply sub in to get:

    1/8 = (arc length)/circumference

    and after cross multiplying:

    circumference = 8(arc length)

    Since we solved for circumference = 12pi, we sub in to get:

    12pi = 8(arc length)

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    adam15 Rising GMAT Star Default Avatar
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    Post Sun Nov 29, 2009 10:51 am
    I was wondering if the sector area is another circle with different radius, do still getting the same result. A is sufficient, because I consider the sector as a unite circle.and i am getting (6/sqr(2))*2.

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    Post Sun Nov 29, 2009 2:56 pm
    adam15 wrote:
    I was wondering if the sector area is another circle with different radius, do still getting the same result. A is sufficient, because I consider the sector as a unite circle.and i am getting (6/sqr(2))*2.
    Think of a circle as a pizza and a sector as a slice.

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