A circle, a square and an octangle walk into a bar
A circle is inscribed into a square of side a = 1 3 . The circle is also inscribed into an octangle. Exactly four octangle's sides are parallel with square's sides, and other four are perpendicular with appropriate square's diagonals, as shown in the image above.
Find the area of the green region.
Give your answer to 3 decimal places.
The answer is 0.909.
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1 solution
Hey @Milan Milanic , I think we must have a psychic connection or something. I just generalized an equation for finding the area between two tangents like two days ago: a t = t a n ( 2 α ) r 2 − 3 6 0 π ⋅ r 2 β Alpha is the angle where the tangents meet (for your example, 135). And beta is of course supplementary to alpha. I got it wrong though, since I am a moron and cannot even use my own equations correctly. Really cool question, it put my idea to the test!
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Hey @Drex Beckman , your comments cheer me up as always. Also, don't be so harsh on yourself (and I am the one who thinks that got the right to tell someone something about self criticism :D ). It really seems that sometimes we have same ideas/thoughts to some extent about some topics (I would say that would be geometry). Since our way of thinking is similar, as I said, maybe we should contact each other on brilliant slack perhaps? :)
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Sure, @Milan Milanic , you seem cool. I am not aware of brilliant slack, though. Is it like a PM thing on here?
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@Drex Beckman – @Drex Beckman I must say that I am not familiar with that PM thing (don't even know what should that be, website, service... no clues).
Here is link to join brilliant slack. It is meant for chatting, discussing about new wikis and so on. But of course you can chat in private with other members.
https://slackin.brilliant.org/
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@Milan Milanic – @Milan Milanic PM is private message. Anyway, I will try to get on. I just have been having Internet access issues for around a month. Can't use all my mobile data. ;)
How MB=MN ?
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Since the problem's statement said that appropriate octangle's sides will be perpendicular, ∠ N M B = 9 0 ° .
∠ M B N = 4 5 ° which is as same as angle formed by square's diagonal and square's side.
The only one left is ∠ M N B which is 4 5 ° due to inner angle sum.
Conclusion, △ N M B is isosceles with M N = M B
I hope that I have not left anything ambigious in my explanation and if I have so, feel free to ask again. :3
Solution:
Square's area is a 2 . Now we will subtract 8 × R e d R e g i o n from square's area.
Since B D is diagonal of square, M B = M N = a 2 2 − 1 .
When we subtract 8 areas of △ M N B , only octagon's area will be left. Then, subtract circle's area and divide it with 8 to get the desired area. 8 2 a 2 ( 2 − 1 − π / 8 ) .
When put in calculator it is 0 . 9 0 8 9 8 6 8 . . . = 0 . 9 0 9