The formula
width = diagonal × ratio ÷ √(width² + height²)
Where it comes from
The diagonal and the aspect ratio form a right-angled triangle: knowing both, Pythagoras gives the two sides. That is why two 55-inch televisions with different ratios have different widths, and why a 34-inch ultrawide is shorter than an ordinary 27.
How to work it out by hand
- Work out the hypotenuse of the ratio: for 16:9 it is √(16²+9²) = 18.36
- Divide the diagonal by that number
- Multiply the result by 16 for the width and by 9 for the height
- Multiply by 2.54 to convert to centimetres
What is worth knowing
For television, the recommended distance is between 1.5 and 2.5 times the diagonal: a 55-inch set looks right between 2.1 and 3.5 metres. In 4K you can sit closer without seeing pixels, and that is a good part of its real advantage. Above about 110 ppi the eye no longer picks out pixels at a normal desk distance.