PPI calculator: pixel density from resolution and diagonal, plus screen width and height, aspect ratio, dot pitch and the distance where pixels blend.
A 27-inch 2,560 × 1,440 screen (16:9) packs 108.8 pixels per inch into a 23.5 × 13.2 in panel; beyond about 32 in (80 cm) a person with 20/20 vision can't pick out single pixels.
At this distance one pixel spans 1 arcminute, the detail 20/20 vision resolves; this is the idea behind “retina” displays.
Pixel density is the diagonal resolution in pixels divided by the diagonal size in inches: PPI = √(w² + h²) ÷ diagonal. The same ratio splits the diagonal into the physical width and height, dot pitch is 25.4 mm ÷ PPI, and the blend distance is where one pixel spans 1 arcminute, the detail that 20/20 vision resolves.
The default 27-inch 2560 × 1440 monitor has 108.79 PPI, a 16:9 panel of 23.53 × 13.24 inches, and pixels that blend beyond about 31.6 inches (80 cm). The same size at 4K, 3840 × 2160, reaches 163.18 PPI and blends beyond 21 inches.
Advertised diagonals are rounded, so results can differ slightly from a spec sheet. The blend distance assumes 20/20 vision; people with sharper eyesight can pick out pixels from further away.
Checked against: Python decimal: √(2560² + 1440²)/27; retina distance 1/(PPI·tan(π/10800))
Checked against: Apple MacBook Air (M1, 2020) tech specs list 227 ppi; Python decimal gives 226.983
Checked against: Python decimal: 1000√2/10
Checked against: Python decimal and math.gcd: gcd(1366, 768) = 2
Find the diagonal in pixels with Pythagoras and divide it by the diagonal in inches. A 27-inch 2560 × 1440 monitor has √(2560² + 1440²) = 2,937.2 pixels along the diagonal, so 2,937.2 ÷ 27 = 108.79 PPI. Use the visible screen diagonal, not the outside of the case.
Apple introduced the name with the iPhone 4 in 2010: a 960 × 640 screen at 326 pixels per inch, where Apple said the eye cannot make out individual pixels at a normal viewing distance. At 326 PPI one pixel spans 1 arcminute from 10.5 inches, so the threshold depends on distance: a 27-inch 4K monitor at 163 PPI passes the same test from 21 inches.
To stop seeing single pixels with 20/20 vision, sit at least about 43 inches (1.1 m) from a 55-inch 4K TV, which has 80.1 PPI, or 51 inches (1.3 m) from a 65-inch one. Beyond about 86 inches (2.2 m), a 55-inch 1080p screen blends its pixels too, so 4K adds little visible detail from that far away.
16:9: divide both numbers by their greatest common divisor, 160. TVs and most monitors are 16:9, while some laptops use 16:10, such as the 13.3-inch 2560 × 1600 MacBook Air (8:5 when reduced). A resolution such as 1366 × 768 reduces only to 683:384, which is within 0.05% of 16:9.
Advertised diagonals are rounded. The iPhone 4's 960 × 640 screen is sold as 3.5 inches, which gives 329.7 PPI, while Apple quotes 326 PPI, matching a diagonal of about 3.54 inches. Enter the diagonal to two decimal places when you know it, and check that the resolution is the panel's native one.
Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “27-inch 2560 × 1440 monitor” is checked against Python decimal: √(2560² + 1440²)/27; retina distance 1/(PPI·tan(π/10800)).
Pixel density — PPI from resolution and diagonal; Visual acuity — 20/20 vision resolves detail of 1 arcminute.
5 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.
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