Trigonometry question, shaded region.?

Find the area of the shaded region in the figure.

http://i575.photobucket.com/albums/ss196/felixduhc...

And how do you get the answer? Explain how you get the answer please. This question kinda confuses me.

3 Answers

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  • 10 years ago
    Favorite Answer

    Do it in bits.

    First find the area of the whole circle...

    πr²=4π

    Then find the area of the sector of the circle you're interested in ...

    The angle at the centre is 120, and this is 1/3 of the whole angle [As 120/360 = 1/3]

    So area sector=4π/3

    Then take off the triangle portion, to leave just the shaded segment...

    Area triangle=(1/2) x (one side) x (another side) x (sine of angle between)

    = 0.5 x 2 x 2 x sin(120)

    = 2 x √3 /2

    =√3

    Subtracting this from your sector,

    Area of segment = (4π/3) - √3

    People often leave answers like that, but you can work it out as a decimal, if you want...

    2.46

    Source(s): Maths teacher
  • 10 years ago

    OK so this is how I'm going to solve the question.

    FIRST, i recognize that 120deg. is one third of 360deg., a whole circle which means i can find a third of the area of the whole circle simply by dividing the area by 3.

    SECOND, I will find the area of the 120deg triangle and subtract that from 1/3 of the area of the circle.

    Two steps. Simple, right?

    Applying the first step.

    A=πr^2

    A=π(2^2)

    A=4π

    A=12.56

    One third of this is 12.56 / 3 = 4.18

    Applying the second step.

    The are of the triangle will equal two times the area of half the triangle, and finding the area of half of that triangle is easy, so that's what we'll do.

    We need first to find the shortest distance between the centre of the circle and the line bounding the shaded area. This distance is 2cos(60deg.) = 1, meaning that, from Pythagoras' theorem, the other side is sqrt(3).

    Now we simply apply the area formula:

    A = 1/2 b*h

    A = 0.5 * sqrt(3)

    A = 0.866

    Multiply this by two and we get the area of the whole triangle.

    =1.73 units^2

    This we now subtract from 4.18 to get the final answer.

    = 4.18 - 1.73

    = 2.44 square units.

    So just recapping...I first found 1/3 of the whole area of the circle, then subtracted the area of the triangle. I found the area of the triangle by splitting it into to two triangles, through the middle so it makes a 90deg angle with the line binding the shaded area. I then used the cosine rule to find the adjacent side, then Pythagoras to the find the other side, then found the area and multiplied it by two.

    Source(s): High School Maths.
  • 10 years ago

    a = 120/360*pi*4 - 1/2*4*sin120

    a= 1/3*4pi-2sin120

    a=4pi/3 - sqrt(3)

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