I am currently designing an webpage which will be used to display dynamic content on an LED screen in a mall.
I need to make sure that the content I design exactly fits the height and width of the display . The screen size is 42″ and the resolution is 1280 × 720 .
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How do I make sure that what I design exactly fits the screen? I
am more worried about making it fit vertically. -
How can I design content which automatically adapts and fits
itself to such long vertical screens? -
Are there any tools that can help me with the design process?

You shouldn’t have many problems, just set your height to 1280px and width to 720px.
A good template would be:
You’ll want to find out what browser they are using, then design in that browser. You may well find that it runs Windows XP Embedded with IE 7.
To make it easy to use IE7, I’d recommend setting up a virtual machine running the browser. You can get this from Microsoft here: http://www.microsoft.com/en-us/download/details.aspx?id=11575&mnui=0, or if you are on OSX or Linux, the frankly amazing https://github.com/xdissent/ievms may be handy.
It would be worth considering carefully what the minimum fontsize is that will be readable. You may also find that if it’s a normal screen that has been rotated that the subpixels will be rotated through 90˚, that is, instead of going RGB left to right, they will be arranged vertically. Microsoft’s Cleartype will look weird in that case, so if that is turned on then you’ll find that smaller text is very blurry. (It works on pixels left and right not up and down, so a rotated screen throws it off considerably).
I’d try to get a copy of the screen, or just run it through your TV and turn your head sideways to approximate it 🙂