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 Fixed length from unit square. (Posted on 2009-10-01)
Given a unit square and a fixed length, r.

Construct the set of all points which are at distance r from some point on the square.

Find the area of this set.

Note: A square is composed of 4 line segments, not the interior. For some values of r the set will have a hole in it.

Example of the construction:

 No Solution Yet Submitted by Jer Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution | Comment 2 of 3 |
`The area of the set is pi*r^2+4*r+1 minus thearea of the center hole.`
`For r < 1/2, `
`   Area of center hole = (1-2*r)^2`
`For 1/2 <= r <= sqrt(2)/2,`
`   No center hole`
`For sqrt(2)/2 < r,`
`   Area of center hole `
`          -- sqrt(r^2 - 1/4)          |          |     =  4 | [sqrt(r^2 - x^2) - 1/2] dx          |          |         -- 1/2  `
`     = 2*r^2*[arctan(t)-arctan(1/t)] - t + 1`
`       where t = 2*sqrt(r^2 - 1/4) and`
`       arctan returns radians.`
` `

 Posted by Bractals on 2009-10-01 14:41:27

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