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    With a homework titled 'Radians', I've got two annoying questions here which I could do with some help with - pointing me in the right direction would be greatly appreciated.
    1. An equilateral triangle is inscribed in a circle of radius 10 cm.
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    (i) Find the area of the circle - Done this.
    (ii) Find the area of the triangle
    (iii)Find the area of the three segments surrounding the triangle. - I know this is relatively easy once I have the area of the triangle (Part ii).

    2. Once upon a time a hermit found an island shaped like a triangle with straightshores of lengths 6 km, 8 km, and 10 km. Needing seclusion, he declared that noone should approach within 1 km of his shore. What was the area of his‘exclusion’ zone?

    Really not sure where to go with the latter, I've drew the triangle and 'exclusion zone' (circle), but I can't think of anything related to radians bar area of a sector. I know to do this I'll need to work out the necessary angles (I'm assuming I use the cosine rule) - but once I have the angles of the triangle, where do I then go?
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    (Original post by Kozmo)
    With a homework titled 'Radians', I've got two annoying questions here which I could do with some help with - pointing me in the right direction would be greatly appreciated.
    1. An equilateral triangle is inscribed in a circle of radius 10 cm.
    Spoiler:
    Show
    (i) Find the area of the circle - Done this.
    (ii) Find the area of the triangle
    (iii)Find the area of the three segments surrounding the triangle. - I know this is relatively easy once I have the area of the triangle (Part ii).

    2. Once upon a time a hermit found an island shaped like a triangle with straightshores of lengths 6 km, 8 km, and 10 km. Needing seclusion, he declared that noone should approach within 1 km of his shore. What was the area of his‘exclusion’ zone?

    Really not sure where to go with the latter, I've drew the triangle and 'exclusion zone' (circle), but I can't think of anything related to radians bar area of a sector. I know to do this I'll need to work out the necessary angles (I'm assuming I use the cosine rule) - but once I have the angles of the triangle, where do I then go?
    no 2 is rather nice ... they should ask it in a Maths interview ...
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    (Original post by Kozmo)
    With a homework titled 'Radians', I've got two annoying questions here which I could do with some help with - pointing me in the right direction would be greatly appreciated.
    1. An equilateral triangle is inscribed in a circle of radius 10 cm.
    Spoiler:
    Show
    (i) Find the area of the circle - Done this.
    (ii) Find the area of the triangle
    You know that the area of a triangle is given by \frac{1}{2}AB \sin c

    You know it's equilateral, so c = 60^{\circ}, both side lengths would be the same as well.
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    (Original post by TeeEm)
    no 2 is rather nice ... they should ask it in a Maths interview ...
    It's not bad is it, made me think so far and I'm yet to even solve it! :')
    Going to give it another attempt tomorrow after 40 winks and see how far I get.

    I'm assuming Cosine and trig is necessary?
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    (Original post by Zacken)
    You know that the area of a triangle is given by \frac{1}{2}AB \sin c

    You know it's equilateral, so c = 60^{\circ}, both side lengths would be the same as well.
    Yep, I managed to get that far. The problem was with working out the side length. I got the side length to be 10√3 - but this required a very long, tedious process and I'm not too sure on it anyway.
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    (Original post by Kozmo)
    It's not bad is it, made me think so far and I'm yet to even solve it! :'
    Going to give it another attempt tomorrow after 40 winks and see how far I get.

    I'm assuming Cosine and trig is necessary?
    An able experienced student should be able to do it without writing much down
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    (Original post by Kozmo)
    Yep, I managed to get that far. The problem was with working out the side length. I got the side length to be 10√3 - but this required a very long, tedious process and I'm not too sure on it anyway.


    Does this help?
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    (Original post by Zacken)


    Does this help?
    Yep. Using that I got the same side value of 10√3 which I got using my very prolonged method! Hopefully this is right, would you possibly be able to support this answer?

    Also, any idea on no. 2?
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    (Original post by TeeEm)
    An able experienced student should be able to do it without writing much down
    I like to consider myself able, although experienced I am not >.>
    I'm quite looking forward to seeing the method though.
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    (Original post by Kozmo)
    Yep. Using that I got the same side value of 10√3 which I got using my very prolonged method! Hopefully this is right, would you possibly be able to support this answer?
    Yep, that's correct.

    Also, any idea on no. 2?
    I'll have a look now.
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    (Original post by Kozmo)

    Really not sure where to go with the latter, I've drew the triangle and 'exclusion zone' (circle), but I can't think of anything related to radians bar area of a sector. I know to do this I'll need to work out the necessary angles (I'm assuming I use the cosine rule) - but once I have the angles of the triangle, where do I then go?
    Don't you just need to find area circle - area triangle? Area triangle should be easy to work out using Heron's.
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    (Original post by Kozmo)
    I like to consider myself able, although experienced I am not >.>
    I'm quite looking forward to seeing the method though.
    I personally can give you the answer without writing anything down ... maybe I have seen this type of problem before ....
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    (Original post by Zacken)
    Don't you just need to find area circle - area triangle? Area triangle should be easy to work out using Heron's.
    You know what, I didn't even consider that - I'm too busy over-complicating it.
    I'll give this a shot, although I have no idea what Heron's formula is!
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    (Original post by TeeEm)
    I personally can give you the answer without writing anything down ... maybe I have seen this type of problem before ....
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    I'm envious :cry2:
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    (Original post by Kozmo)
    You know what, I didn't even consider that - I'm too busy over-complicating it.
    I'll give this a shot, although I have no idea what Heron's formula is!
    Don't bother using Heron, bit of a fancy shmoozle, stick with what you do know.
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    (Original post by Kozmo)
    I'm envious :cry2:
    if I understood the problem well i get 24 + pi
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    (Original post by Zacken)
    Don't bother using Heron, bit of a fancy shmoozle, stick with what you do know.
    I just encountered a problem: obviously I have to work out the area of the circle, but how do I go about this if I don't know the radius?
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    (Original post by Kozmo)
    I just encountered a problem: obviously I have to work out the area of the circle, but how do I go about this if I don't know the radius?
    Ah, nevermind. The method you're meant to use is to notice that the excluded region consists of a rectangular region (1 km wide) across each shore and a circular region at the vertices. Picture:



    That should do the trick.
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    (Original post by Zacken)
    Ah, nevermind. The method you're meant to use is to notice that the excluded region consists of a rectangular region (1 km wide) across each shore and a circular region at the vertices. Picture:



    That should do the trick.
    Ahh, I see. I'll give this a go tomorrow. Thank you very much for your help!

    Oh and I just realised another, much easier way to solve the first one so thought I'd share:

    If we split the equilateral up into a further three triangles the inside angles are all 120 degrees, and thus we can use the area of a segment formula = 0.5xr^2(theta-sintheta) to calculate the segment area. Multiply this by three and minus it from the area of the circle.
    - Same result but much more efficient method in my opinion!

    Anyhow, I'm going to head to sleep.
    Thanks to anyone who helped!
 
 
 
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