 ## Thursday, September 17, 2009

We have finally decided to buy a flat screen TV. My husband thought that our current 27" box does not allow him to see all the details of the recent, tense and thrilling, football game. He is considering a giant 54" screen. Will this screen increase our viewing area by two?

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Alin Grin said...

Since the TV sizes are the length of the diagonal line which stretches between the opposite corners of the viewing area. By buying a 54'' TV to replace the 27'' TV the viewing area will increase by 4. Intuitively thinking you will need 4 27'' TVs in order to create a 54'' two diagonal lines across the viewing area.
You can also use Pythagoras's a^2+b^2=c^2 to calculate sizes of the TVs which are not as intuitive as in this case.

Maria said...

Here is what MarkinKentuckiana posted on the Open Salon feed of this puzzle:

No.

Being too lazy to actually check, I think doubling the diagonal would roughly square the area. No way to be certain without knowing the horizontal and vertical dimensions of both - and what allowances are we making for that portion of each screen covered by the bezel.

Both, he and Alin are right about approximately squaring the size of the viewing area to around 4. Markin brings up a good point, that Alin and I missed: Resolution of the TVs may be different and in fact it is. My old box is has a standard 4:3 screen when the new flat screen I assume will be HDTV (resolution 16:9). If resolution would be the same, then Alin's great example of stacking 4 small TVs in a 2x2 arrangement to create larger one with double size diagonal will apply and the viewing area will increase by 4. Since resolution is different, area ratio will be a different number. And I think we can actually compute it.... Even without knowing the width and height.
Who knows how?

Maria said...

By the way, here is a fun story I wrote about HDTV:
How Patrick Dempsey and HDTV may be responsible for obesity

Maria said...

The giant TV is here and we watched our first feature film on it, a thriller that left us all shivering.
Looking at the new and old TVs, I can see that the new one is definitely not four times larger.
Here is the math:

my old TV is 4:3 resolution, 27" diagonal
my new TV is 16:9 resolution (HDTV), 54" diagonal

Assume that the sides of the old TV are 3x and 4x. Then we have: (3x)^2 + (4x)^2 = 27^2
9x^2 + 16x^2 = 25x^2 = 27^2 = 729
from this: x^2 = 729 / 25
Area of the small TV is: 3x times 4x = 12 x^2
So, its area is 12 times 729 / 25 = 350

Now, assume the sides of the new TV are 9y and 16y. Then we have: (9y)^2 + (16y)^2 = 54^2
81y^2 + 256y^2 = 337y^2 = 54y^2 = 2916
from this: y^2 = 2916 / 337
Area of the new TV is: 9y times 16y = 144y^2
Substituting 2916 / 337 for y^2 we get:
144 times 2916 / 337 = 1246

So, viewing area of the old TV is 350 square inch. Viewing area of the new TV is 1246 square inch, approximately 3.56 times larger.

Turns out you could actually compute the size from the diagonal and TV resolution. Perhaps this is why diagonal value is being used in the ads and stores to describe TV size, instead of the usual width and height parameters that are used to define the size of rectangle.