What PPI tells you, and what it does not

Pixels per inch is one of the few display numbers that is honest arithmetic all the way down: two pixel counts, one length, and Pythagoras in between. It is also routinely asked to answer a question it does not contain the inputs for.

The method

A screen's resolution gives you a rectangle of pixels. The diagonal of that rectangle, measured in pixels, is √(width² + height²). The panel's diagonal, measured in inches, is the number on the box. Divide one by the other and you have pixels per inch:

• PPI = √(width px² + height px²) / diagonal in

Take 2560 by 1440 on a 27-inch diagonal. Then 2560² + 1440² = 8627200, and √8627200 ≈ 2937.209560 pixels along the diagonal. Divide: 2937.209560 / 27 ≈ 108.785539 PPI. That is exactly what the free Screen PPI calculator prints for its default inputs, rounded to six decimals.

The reciprocal is often the more useful figure. One pixel is 1 / 108.785539 ≈ 0.009192 inches across — the pixel pitch.

The assumption hiding in the square root

Combining the two pixel counts into one diagonal quietly assumes the pixels are square: that a step of one pixel across covers the same physical distance as a step of one pixel down. Check it on the example. A 16:9 panel with a 27-inch diagonal is 23.532540 by 13.237053 inches (rounded). Horizontally, 2560 / 23.532540 ≈ 108.785539. Vertically, 1440 / 13.237053 ≈ 108.785539. Identical, because the pixel grid's ratio and the panel's ratio are the same 16:9.

When those two ratios disagree — a 1280 by 1024 grid, which is 5:4, driven onto a 16:9 panel — the pixels are not square, the horizontal and vertical densities differ, and the single PPI number is a blend of two different figures rather than either of them. The formula still returns something; it just stops describing one physical quantity.

Three panels, and the number that changes everything

Run three sets of inputs through the same formula:

• 2560 × 1440 on 27 inches: 108.785539 PPI
• 1920 × 1080 on 15.6 inches: 141.211998 PPI
• 1080 × 2400 on 6.1 inches: 431.443519 PPI
• 3840 × 2160 on 65 inches: 67.781759 PPI

The spread is enormous: 431.443519 / 108.785539 ≈ 3.966001, so the handheld panel packs almost four times the pixels into an inch that the desk panel does. Read as a ranking of anything a viewer experiences, that is meaningless, because PPI is measured per inch of panel, and what reaches an eye is measured per degree of view. Those are different denominators, and the thing that converts between them is viewing distance — which the formula does not contain and the tool does not ask for.

Deriving the conversion

A pixel is 1 / PPI inches across. Held at a distance D inches, it spans an angle of about 1 / (PPI × D) radians, from the small-angle relationship that a short length divided by its distance approximates the angle it subtends. One radian is 180 / π degrees, so the number of pixels that fit into one degree is:

• pixels per degree ≈ PPI × D × π / 180, where π / 180 ≈ 0.017453293

Now put the three panels at plausible distances and recompute. The handheld panel at 12 inches: 431.443519 × 12 × 0.017453293 ≈ 90.361319 pixels per degree. The desk panel at 24 inches: 108.785539 × 24 × 0.017453293 ≈ 45.567980. The large panel at 96 inches: 67.781759 × 96 × 0.017453293 ≈ 113.569427.

The order inverts. The panel with the lowest PPI of the three delivers the most pixels per degree at those distances — 113.569427 / 90.361319 ≈ 1.256837 times as many as the handheld one, despite having under a sixth of its PPI. Change the distances and the ranking changes again, which is the entire point: PPI is a property of the panel, pixels per degree is a property of the panel and where you are.

Two honest caveats. The small-angle step is an approximation; the exact angle a pixel subtends is 2 × arctan(1 / (2 × PPI × D)), and at these pixel sizes the difference is far below the precision anyone quotes. And this page will not tell you what any of it looks like — whether a difference in pixels per degree is noticeable depends on the viewer, the content and the lighting, none of which are numbers you can get from a resolution and a diagonal.

Using the tool honestly

The calculator takes three inputs — width in pixels, height in pixels, diagonal in inches — and returns one number. Feed it the panel's active diagonal rather than a bezel-to-bezel measurement, keep the ratios in mind if the grid and the panel disagree, and treat the result as what it is: the pixel density of a surface, not a verdict about a picture.