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# C2 Questions watch

1. 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?
2. (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 ...
3. (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

You know it's equilateral, so , both side lengths would be the same as well.
4. (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?
5. (Original post by Zacken)
You know that the area of a triangle is given by

You know it's equilateral, so , 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.
6. (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
7. (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?
8. (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?
9. (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.
10. (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.
11. (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.
12. (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 ....
13. (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!
14. (Original post by TeeEm)
I personally can give you the answer without writing anything down ... maybe I have seen this type of problem before ....
I'm envious
15. (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.
16. (Original post by Kozmo)
I'm envious
if I understood the problem well i get 24 + pi
17. (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?
18. (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.
19. (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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