Apple’s Retina Display marketing broadly publicized the concept of retinal acuity, but each person’s vision differs; so, just how small do those pixels need to be for your vision?

Fortunately, inverting the well known Snellen notation (e.g. 20/20 corrected vision, 20/30 uncorrected vision, etc…) gives your personal visual acuity in minutes of arc. For example, inverting 20/20 = 1 meaning that 20/20 vision can resolve 1 arc minute sized details. Similarly, someone with 20/60 vision has a visual acuity of 60/20 = 3.3 arc minutes; 20/15 vision can resolve 15/20 = 0.75 arc minutes. Go ahead and calculate your own visual acuity in arc minutes. Ready?

OK, let’s see how tiny the pixels on a screen need to be to make it a retina display *for you*. To do this, we’ll calculate the smallest pixels that you can resolve at a given distance. For example, if you have 20/20, or *1 arc minute*, vision and hold a smartphone 11 inches (28 cm) away, you’ll be able to resolve individual pixels if there are 313 pixels per inch (123 pixels/cm) or less; if the actual pixels are smaller (i.e. higher pixel density), then it’s a “retina display”.

Here’s how to calculate this:

tan(½ × *1 arc minute*) × 2 × 11 inches = 0.0032 inches (or the inverse of 313 pixels per inch (ppi) or more)

tan(½ ×* 1 arc minute*) × 2 × 28 cm = 0.00814 cm (or 123 pixels per cm (ppcm) or more)

Spreadsheet formulas for this looks like:

*<resolvable pixel>*` = tan(radians(0.5 * `

*<your arc min.>*`/60)) * 2 * `

*<distance>*

*<pixel density>*` = 1 / `

*<resolvable pixel>*

In more detail: to calculate the pixel size, `s`

, opposite the viewer divide the angle, `a`

, in half to give a right triangle with the viewing distance, `d`

, adjacent to the angle and the length of ½ of a pixel opposite. 1 arc minute = ^{1}/_{60} degree. Then with basic trigonometry:

tangent (angle) = ^{opposite}/_{adjacent}

tangent (½ `a`

) = ^{½ s}/_{d}

½ `s`

= tan( ½ `a`

) `d`

`s`

= tan( ½ `a`

) 2 `d`

If you were looking at a television 5-½ feet away instead, then you’d only be able to resolve 52 ppi (20 ppcm):

tan(½ ×* 1 arc minute*) × 2 × 66 in. = 0.0192 in. or 52 ppi

tan(½ ×* 1 arc minute*) × 2 × 170 cm = 0.0495 cm or 20 ppcm

A 42-inch diagonal, full HD television (1920×1080) also happens to have 52 pixels per inch; therefore, when viewed from 5-½ feet or farther the pixels begin to blur together for 20/20 vision. Homework: how close/far should you sit from your television to turn it into a “retina” display? Enjoy!

Snellen acuity |
Visual resolution (arc minutes) |
Retina display, iPhone |
Retina display, TV |

(11 in, ppi) |
(28cm, ppcm) |
(9′, ppi) |
(2.75m, ppcm) |

20/200 |
10 |
31 |
12 |
3 |
1 |

20/100 |
5 |
63 |
25 |
6 |
3 |

20/70 |
3.5 |
89 |
35 |
9 |
4 |

20/50 |
2.5 |
125 |
49 |
13 |
5 |

20/30 |
1.5 |
208 |
82 |
21 |
8 |

20/20 |
1 |
313 |
123 |
32 |
13 |

20/15 |
0.75 |
417 |
164 |
42 |
17 |