Showing the workings

Over on Register Hardware, they’ve just published a piece I wrote about how far away from an HD TV you should be sitting. The answer is fairly simple, and not really much of a surprise, but we decided to start with the figures from some BBC research and work it out for ourselves, from first principles.

There are a few figures in the feature, and I thought that (since I was taught at O level maths) it would be a good idea to show how we got there.

First up, I said the height of a screen, H, is given using this formula:

How did I get there? Pythagoras. If the screen is a standard 16:9 ratio, and we call D the diagonal, then the screen is 9x high and 16x wide. Thanks to the Greeks we know that


Which we can rewrite as:


And we already know that the height, H, is 9x, which gives us the original formula.

The next bit required some trigonometry, and my memory was jogged by someone on the office who did maths a lot more recently than me, with the phrase ‘SOHCAHTOA.’

We know (from the BBC research, and other sources) that the eye can see detail that covers 1 minute of arc, or 1/60th of a degree. Imagine a triangle touching the screen, one pixel high, with the eye a distance d away, along the middle axis of the triangle.

I’d draw a diagram, but I’m rubbish at those. We can make this into two right angle triangles, dividing along that axis, so now the height is half a pixel, the side at right angles to the screen is d, and the angle at the other end is 1/120th of a degree.

SOHCAHTOA tells us that the tangent of the angle is the ratio of the opposite side (half a pixel height) to the adjacent side (the distance from screen to eye), so we can say


We know that the pixel height is 1/1080th of the screen height H, so we can say the distance at which you can view the screen, and a pixel will cover 1 minute of arc at your eyeball (d) is

We’ve already worked out what H is equal to, further up the page, and tan(1/120) is a constant too. So, working all that out, we can substitute the values to come up with

And to allow for people starting with a screen size in inches, and wanting a distance in metres, we use the factor 0.0254, which results in this formula for the viewing distance (d, in metres) from a particular screen size (D, in inches):

Swapping that round, to start with the viewing distance and find out the screen size you need gives the final formula in the article:

You can, as one of the readers has pointed out, simplify this a bit further if you want, but then it wouldn’t look so scientific, would it?

7 Replies to “Showing the workings”

  1. Really, some teacher thought SOHCAHTOA would be an easy acronym to remember ? One of my teachers from decades ago taught us :
    Our Athletic Team, Often Have Shields, And Hold Cups

    Much easier to remember !

  2. It’s a volcano, apparently. Or perhaps an island with a volcano. Once someone mentioned it in the office, it all came flooding back. But probably a quarter century since I did anything like that

  3. Doh, I thought the recommended distance was no more than double the screen diagonal ?
    This would result in cheapskates with 32″ sets watching TV whilst sitting on their coffee tables !

    1. All the recommendations seem to be based on image height – I guess that’s because it’s easier to cover both traditional (4:3) and widescreen sets that way.

      The ITU (International Telecommunications Union) recommends 3 time screen height for HD, and around 5 or 6 times for standard def. And yes, you’re right, in many cases for an optimum experience, people should really be sitting much closer than they appear to be.

  4. Not simplifying formulae (and indeed, using degrees not radians), just looks amateurish. It’s pre-GCSE maths.

  5. SOHCAHTOA still floats around in my head, decades later. Wish my teachers could have taught me my g/f’s birthday.

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