Think in viewing angle
A screen's apparent size is the horizontal angle it fills in your view. A 55-inch TV at 6 feet and a 100-inch TV at 11 feet look almost the same size, because they fill nearly the same angle. SMPTE's cinema guideline EG 18-1994 gives the formula and a reference point. The formula for the horizontal field of vision is:
The reference point: SMPTE wrote that experience indicates a screen image appears small if it fills less than about 30 degrees of the horizontal field of vision. The guideline was written for movie theaters and withdrawn in 2003, but 30 degrees is still a useful figure for the farthest comfortable seat. The tables also use 36 and 40 degrees. These are simply larger angles for a more immersive picture, closer to the screen; pick whichever suits the room and how you watch.
Width and height from the diagonal
TVs are sold by diagonal, but the angle depends on width. For a 16:9 screen the diagonal is the hypotenuse of a right triangle with sides in the ratio 16 to 9, so the Pythagorean theorem calculator gives the sides: 16² + 9² = 337, and √337 is 18.358.
height = diagonal × 9 ÷ 18.358 ≈ diagonal × 0.490
For other shapes, such as an ultrawide 21:9 monitor, the aspect ratio calculator works out width and height from any ratio.
Viewing distance chart
Distance from your eyes to the screen at which a 16:9 screen fills 30, 36 and 40 degrees of your horizontal view. The 30 degree column is the farthest seat before the picture starts to look small; the 40 degree column is the closest of the three.
| TV | Width × height | 30° | 36° | 40° |
|---|---|---|---|---|
| 32 in | 27.9 × 15.7 in | 4.3 ft | 3.6 ft | 3.2 ft |
| 40 in | 34.9 × 19.6 in | 5.4 ft | 4.5 ft | 4.0 ft |
| 43 in | 37.5 × 21.1 in | 5.8 ft | 4.8 ft | 4.3 ft |
| 50 in | 43.6 × 24.5 in | 6.8 ft | 5.6 ft | 5.0 ft |
| 55 in | 47.9 × 27.0 in | 7.5 ft | 6.1 ft | 5.5 ft |
| 65 in | 56.7 × 31.9 in | 8.8 ft | 7.3 ft | 6.5 ft |
| 75 in | 65.4 × 36.8 in | 10.2 ft | 8.4 ft | 7.5 ft |
| 77 in | 67.1 × 37.8 in | 10.4 ft | 8.6 ft | 7.7 ft |
| 85 in | 74.1 × 41.7 in | 11.5 ft | 9.5 ft | 8.5 ft |
| 98 in | 85.4 × 48.0 in | 13.3 ft | 11.0 ft | 9.8 ft |
| 100 in | 87.2 × 49.0 in | 13.6 ft | 11.2 ft | 10.0 ft |
The same distances in meters:
| TV | Width × height | 30° | 36° | 40° |
|---|---|---|---|---|
| 32 in | 71 × 40 cm | 1.3 m | 1.1 m | 1.0 m |
| 40 in | 89 × 50 cm | 1.7 m | 1.4 m | 1.2 m |
| 43 in | 95 × 54 cm | 1.8 m | 1.5 m | 1.3 m |
| 50 in | 111 × 62 cm | 2.1 m | 1.7 m | 1.5 m |
| 55 in | 122 × 68 cm | 2.3 m | 1.9 m | 1.7 m |
| 65 in | 144 × 81 cm | 2.7 m | 2.2 m | 2.0 m |
| 75 in | 166 × 93 cm | 3.1 m | 2.6 m | 2.3 m |
| 77 in | 170 × 96 cm | 3.2 m | 2.6 m | 2.3 m |
| 85 in | 188 × 106 cm | 3.5 m | 2.9 m | 2.6 m |
| 98 in | 217 × 122 cm | 4.0 m | 3.3 m | 3.0 m |
| 100 in | 221 × 125 cm | 4.1 m | 3.4 m | 3.0 m |
Worked example: a 65-inch screen is 56.7 inches wide. Half of that is 28.3 inches, and tan 15° is 0.268, so the 30 degree distance is 28.3 ÷ 0.268 = 106 inches, or 8.8 feet. The rule of thumb in the chart: the 30 degree distance is about 1.87 times the screen width, the 36 degree distance 1.54 times and the 40 degree distance 1.37 times.
From your seat to a screen size
Measure from where your eyes are on the sofa to the wall or stand where the screen will be; the length converter switches between feet, inches and meters. Then read the diagonal that fills each angle:
| Seat distance | 30° | 36° | 40° |
|---|---|---|---|
| 6 ft (1.8 m) | 44 in | 54 in | 60 in |
| 7 ft (2.1 m) | 52 in | 63 in | 70 in |
| 8 ft (2.4 m) | 59 in | 72 in | 80 in |
| 9 ft (2.7 m) | 66 in | 81 in | 90 in |
| 10 ft (3.0 m) | 74 in | 89 in | 100 in |
| 12 ft (3.7 m) | 89 in | 107 in | 120 in |
| 14 ft (4.3 m) | 103 in | 125 in | 140 in |
Screens come in fixed sizes, so round to the nearest one you can buy. At 9 feet the 30 degree screen is 66 inches and the 36 degree screen 81 inches, so a 65-inch set sits right at the 30 degree line and a 75-inch set falls between the two.
A room with more than one seat
Most living rooms have a sofa and a chair at different distances, and one screen has to serve both. SMPTE's cinema guideline handled the same problem by sizing the image for the most distant viewer, so start there. Size the screen so the farthest seat still gets 30 degrees, then check the nearest seat against the closer columns.
Example: a sofa at 12 feet and an armchair at 8 feet. The far seat needs about 89 inches for 30 degrees. With an 85-inch set, the sofa sees 29 degrees and the armchair 42 degrees. The near seat is then a little past 40 degrees, which some people enjoy and others find too much head turning. A 75-inch set gives 26 and 38 degrees: a little small from the sofa, comfortable from the chair. If the two seats are far apart, no single size hits the same angle for both, and the choice comes down to which seat gets used most.
Measure to the eyes, not to the front of the sofa. Someone leaning back sits a foot or more behind the seat edge, and at these distances one foot moves the angle by a few degrees.
When 1080p and 4K stop making a difference
The limit of what the eye can resolve sets a second distance. EyeWiki, the American Academy of Ophthalmology's reference, puts the minimum angle of resolution of an ideal eye under optimal conditions at about 1 minute of arc, a sixtieth of a degree, and that is the detail size behind 20/20 on an eye chart. A pixel that spans less than 1 arcminute is too small to make out on its own. The distance where one pixel spans exactly 1 arcminute is:
A 1080p screen has 1,920 pixels across, a 4K screen 3,840. Closer than the distance in the table, a 20/20 eye can in principle pick out the pixel structure; farther away, it cannot, and the extra pixels add little that you can see.
| TV | 1080p | 4K |
|---|---|---|
| 32 in | 4.2 ft (1.3 m) | 2.1 ft (0.6 m) |
| 40 in | 5.2 ft (1.6 m) | 2.6 ft (0.8 m) |
| 43 in | 5.6 ft (1.7 m) | 2.8 ft (0.9 m) |
| 50 in | 6.5 ft (2.0 m) | 3.3 ft (1.0 m) |
| 55 in | 7.2 ft (2.2 m) | 3.6 ft (1.1 m) |
| 65 in | 8.5 ft (2.6 m) | 4.2 ft (1.3 m) |
| 75 in | 9.8 ft (3.0 m) | 4.9 ft (1.5 m) |
| 77 in | 10.0 ft (3.1 m) | 5.0 ft (1.5 m) |
| 85 in | 11.1 ft (3.4 m) | 5.5 ft (1.7 m) |
| 98 in | 12.7 ft (3.9 m) | 6.4 ft (1.9 m) |
| 100 in | 13.0 ft (4.0 m) | 6.5 ft (2.0 m) |
Put the two charts together. At the 30 degree distance, one 1080p pixel spans about 0.96 arcminutes at the screen center, already at the limit, so 4K adds little from there. At 36 degrees a 1080p pixel spans 1.16 arcminutes and at 40 degrees 1.30, while a 4K pixel stays at 0.65 or less. So 4K pays off when you sit close enough for a large viewing angle, which is the reason large screens and close seats go together. The guide to PPI vs DPI covers the same pixel density idea for phones, monitors and prints.
What the chart leaves out
- Vision varies. 1 arcminute is the figure for an ideal eye under optimal conditions. Many people see less detail, a few see more.
- Source quality. A compressed stream at 4K can show less real detail than its pixel count suggests.
- Mounting height. The chart is about horizontal angle. How high the screen hangs is a separate question.
- The room. Walkways, doors and windows often fix the seat distance before any chart does. Start from the distance you have, then pick the size.
Frequently asked questions
How far should I sit from a 65-inch TV?
A 65-inch 16:9 screen is 56.7 inches wide. It fills 30 degrees of your horizontal view from 8.8 feet, 36 degrees from 7.3 feet and 40 degrees from 6.5 feet. Any seat between 6.5 and 8.8 feet puts the screen between 40 and 30 degrees.
What size TV do I need for a 10-foot viewing distance?
At 10 feet, a 16:9 screen fills 30 degrees at a diagonal of about 74 inches, 36 degrees at 89 inches and 40 degrees at 100 inches. Anything smaller than the first figure will look small from that seat.
Can I see the difference between 1080p and 4K?
It depends on distance. A 20/20 eye resolves detail down to about 1 arcminute. On a 65-inch screen a 1080p pixel shrinks below that from about 8.5 feet, a 4K pixel from about 4.2 feet. Sitting between those distances, 4K can show detail 1080p cannot; farther back than the 1080p figure, both resolutions are finer than that limit.
Where does the 30 degree figure come from?
SMPTE engineering guideline EG 18-1994, written for cinema design, says experience indicates a screen image appears small if it occupies less than about 30 degrees of the horizontal field of vision. SMPTE withdrew the guideline in 2003, so treat 30 degrees as a long-used reference for the farthest seat, not a current standard.
How do I find the width of my TV from the diagonal?
For 16:9, multiply the diagonal by 0.872 for the width and by 0.490 for the height. Both follow from the Pythagorean theorem: the diagonal is the hypotenuse of a triangle whose sides are in the ratio 16 to 9.
Sources
- SMPTE Engineering Guideline EG 18-1994, Design of Effective Cine Theaters (an image appears small below about 30 degrees of the horizontal field of vision; horizontal field of vision = 2 × arctan(½W ÷ D); withdrawn March 26, 2003)
- EyeWiki, American Academy of Ophthalmology: Visual Acuity Testing in Adults (minimum angle of resolution about 1 minute of arc for an ideal eye; Snellen letters of 5 arcmin with 1 arcmin elements; 20/20 as the norm)