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        riddles >> putnam exam (pure math) >> Semi-circle in a Square
        
(Message started by: THUDandBLUNDER on Dec 15^th, 2006, 6:28pm)

Title: Semi-circle in a Square
Post by THUDandBLUNDER on Dec 15^th, 2006, 6:28pm

What is the largest semi-circle you can fit inside a unit square?

Title: Re: Semi-circle in a Square
Post by flamingdragon on Dec 15^th, 2006, 10:07pm

A semi-circle the size of a unit square. ::)

Title: Re: Semi-circle in a Square
Post by Barukh on Dec 16^th, 2006, 5:14am

[hide]0.58[/hide]?

Title: Re: Semi-circle in a Square
Post by Whiskey Tango Foxtrot on Dec 16^th, 2006, 10:14am

Are you talking about 1.414/2 Barukh?

Title: Re: Semi-circle in a Square
Post by Grimbal on Dec 16^th, 2006, 10:36am

[hide]looks more like 2-sqrt(2), as a radius[/hide]

Title: Re: Semi-circle in a Square
Post by Whiskey Tango Foxtrot on Dec 16^th, 2006, 10:54am

Of course. Thanks G-man.

Title: Re: Semi-circle in a Square
Post by flamingdragon on Dec 16^th, 2006, 10:58am

I take it then that I was wrong?

Title: Re: Semi-circle in a Square
Post by Ken_Wiley on Dec 16^th, 2006, 11:32am

Does seem like would be bigger than just half of a circle in it.

Title: Re: Semi-circle in a Square
Post by Eigenray on Dec 16^th, 2006, 1:47pm

on 12/16/06 at 10:36:56, Grimbal wrote:

as a radius

Ahhh, I had gotten [hide].539[/hide], and thought, "well, that's less than Barukh's answer, so I must be missing something..."

Title: Re: Semi-circle in a Square
Post by Barukh on Dec 17^th, 2006, 12:39am

A follow up: what is the largest circular sector with central angle http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif that can be fit in a unit square?

Title: Re: Semi-circle in a Square
Post by Grimbal on Dec 17^th, 2006, 6:21am

Is alpha given?

[hide] Then I'd say you have to consider 3 cases [/hide]

Title: Re: Semi-circle in a Square
Post by Barukh on Dec 18^th, 2006, 11:27pm

on 12/17/06 at 06:21:55, Grimbal wrote:

Is alpha given?

Yes, of course.

Quote:

[hide] Then I'd say you have to consider 3 cases [/hide]

Why [hide]3[/hide]?

Title: Re: Semi-circle in a Square
Post by Grimbal on Dec 19^th, 2006, 1:32am

[hide]There would be one formula for alpha<1/2·pi,
one formula from 1/2·pi/2 to 3/2·pi
and a third for alfa>3/2·pi

I am just a little lazy to figure out the formulas. I think the last one is the constant 1/2 if the radius is what is asked.[/hide]