Formula
Width is the diagonal times the width part of the aspect, divided by the hypotenuse of the aspect. Height uses the height part. A viewing distance is the distance at which that width fills a chosen horizontal angle: distance equals width divided by twice the tangent of half the angle. The same identity run backwards turns a seat into a diagonal.
W = diag × a / √(a² + b²)
H = diag × b / √(a² + b²)
D = W / (2 tan(θ/2))
θ = 2 atan(W / (2 D))
How a TV distance calculator uses screen width
The angle is what your eyes actually fill. A wider screen, or a wider aspect of the same diagonal, reaches a given angle from farther away. On a 16:9 screen the 40° distance is about 1.1973 times the diagonal, the 36° distance about 1.3412 times the diagonal, and the 30° distance about 1.6264 times the diagonal. Those shortcuts are only for 16:9. A 21:9, 64:27, or 4:3 screen uses the width.
THX, SMPTE, and why there is a range
THX treats about 40° as the recommended, immersive field of view, and about 36° as the maximum for a back row. SMPTE’s recommendation, cited here from SMPTE EP 196, is at least a 30° horizontal field of view, which this page labels relaxed mixed use. There is no single correct seat. The recommended range runs from the 40° distance to the 30° distance, and the 36° point is marked inside it. A cinema and a living room are not the same room. Personal preference matters, and these figures are guidelines.
Resolution, 20/20 acuity, and a TV size calculator
A 20/20 Snellen eye resolves about one arcminute. Pixel pitch is the screen height divided by the vertical rows: 720, 1,080, 2,160 for 4K, or 4,320 for 8K. The acuity distance is that pitch divided by the tangent of one arcminute, about 3,437.75 times the pitch. Sit at that distance or closer to see the full detail of the format. Farther away, the extra resolution is finer than 20/20 vision and is not visible. 4K has twice the rows of 1080p, so its acuity distance is half as far, which is why 4K lets you sit closer before the pixels show.
p = H / rows
Dacuity = p / tan(1/60°)
H = D × tan(1/60°) × rows
Choosing a TV size for a room starts from the seat, not from the showroom. The 40° diagonal is the largest recommended, immersive size. The 30° diagonal is the smaller, relaxed size. Common sizes in between are 43, 50, 55, 65, 75, 77, 83, 85, 98, and 100 inches, and this page keeps the ones that land in the range. The acuity check then gives the smallest screen at which the chosen resolution is fully resolvable. At a normal living-room distance that 4K screen is often larger than anything in the range. The sets you can buy still look sharp. You simply are not resolving every 4K pixel, and that is normal.
TV height calculator, eye level, and the fireplace
The comfortable mount puts the centre of the screen at seated eye height. A sofa is often about 42 inches, roughly 107 cm, and a typical range is about 38 to 44 inches. The bottom edge is one half of the image height below that centre, and the top edge is one half above it. If the centre is higher, over a fireplace or a tall console, the upward angle is the arctangent of that rise divided by the seat distance. Under about 10 to 15° is still a small angle. Above that, a tilting mount or a lower centre is more comfortable. This is comfort guidance only. Follow the mount and TV manufacturer instructions and the VESA pattern. This page does not say whether a wall or a stud can carry the set, and it does not account for heat above a fireplace.
centre = eye height
bottom = eye − H/2
top = eye + H/2
up = atan((centre − eye) / D)
The same angles apply to a projector screen. Throw distance, the lens, and the brightness are a different calculation, on the projector throw distance calculator. The enclosure in the room is the subwoofer box calculator.
Worked examples
65 in, 16:9. Width 56.65 in, height 31.87 in. 40° 6.49 ft, 36° 7.26 ft, 30° 8.81 ft. 4K acuity 4.23 ft, 1080p acuity 8.45 ft.
55 in, 16:9. Width 47.94 in, height 26.96 in. 40° 5.49 ft, 36° 6.15 ft, 30° 7.45 ft. 4K 3.58 ft, 1080p 7.15 ft.
75 in, 16:9. Width 65.37 in, height 36.77 in. 40° 7.48 ft, 36° 8.38 ft, 30° 10.16 ft. 4K 4.88 ft, 1080p 9.75 ft.
85 in, 16:9. Width 74.08 in, height 41.67 in. 40° 8.48 ft, 36° 9.50 ft, 30° 11.52 ft. 4K 5.53 ft, 1080p 11.05 ft.
10 ft seat, 16:9. 40° size 100.22 in, 30° size 73.78 in. Common sizes 75, 77, 83, 85, 98, 100 in. 4K fully resolvable from 153.79 in up.
65 in at 8 ft. The screen fills 32.88°.
65 in, eye height 42 in. Centre 42.00 in, bottom edge 26.07 in, top edge 57.93 in.
Centre at 60 in, eye at 40 in, seat at 10 ft. Upward angle 9.46°.
Reference
The eight checks for this page. Distances are seat to screen. Heights are above the floor. 16:9 unless noted.
| Case | Result |
|---|---|
| 65 in | 56.65 × 31.87 in · 40° 6.49 ft · 36° 7.26 ft · 30° 8.81 ft · 4K 4.23 ft |
| 55 in | 47.94 × 26.96 in · 40° 5.49 ft · 30° 7.45 ft · 4K 3.58 ft |
| 75 in | 65.37 × 36.77 in · 40° 7.48 ft · 30° 10.16 ft · 4K 4.88 ft |
| 85 in | 74.08 × 41.67 in · 40° 8.48 ft · 30° 11.52 ft · 4K 5.53 ft |
| 10 ft, size | 40° 100.22 in · 30° 73.78 in · 75, 77, 83, 85, 98, 100 in · 4K from 153.79 in |
| 65 in at 8 ft | 32.88° |
| 65 in, eye 42 in | centre 42.00 in · bottom 26.07 in · top 57.93 in |
| Centre 60 in, eye 40 in, 10 ft | 9.46° up |
Questions
How far should I sit from a 65-inch TV?
A 65-inch 16:9 screen is 56.65 inches wide and 31.87 inches tall. The immersive seat, a 40° horizontal angle, is 6.49 feet. The THX back row at 36° is 7.26 feet. The relaxed SMPTE 30° seat is 8.81 feet. Sit in that span. At 4K the 20/20 acuity distance is 4.23 feet, and at 1080p it is 8.45 feet.
What size TV for a 10-foot distance?
At 10 feet a 16:9 screen that fills 40° is a 100.22-inch diagonal. A 30° screen is 73.78 inches. Common sizes in that range are 75, 77, 83, 85, 98, 100 inches. 4K is fully resolvable from 153.79 inches up at that seat, so a set in the recommended range looks sharp and does not show every 4K pixel. That is normal.
Is 4K worth it at my distance?
4K has twice the vertical pixels of 1080p, so the acuity distance is half as far: you can sit closer before the pixels become visible. Farther than that distance, the extra pixels are finer than 20/20 vision resolves. The picture still looks sharp. Whether 4K is worth it depends on how close you sit, and on whether you want that detail when you do.
How high should a TV be mounted?
Put the centre of the screen at seated eye height. A typical sofa is about 42 inches, roughly 107 cm, and most sofas fall around 38 to 44 inches. On a 65-inch 16:9 set that puts the centre at 42.00 inches, the bottom edge at 26.07 inches, and the top edge at 57.93 inches. Follow the mount and the TV maker’s instructions, including the VESA pattern. This is not structural guidance.
Is it bad to mount a TV above the fireplace?
It raises the centre above eye level, so you look up. A centre at 60 inches, eyes at 40 inches, and a seat at 10 feet is a 9.46° upward angle, still under about 10 to 15°. Steeper than that is less comfortable. A tilting mount or a lower centre helps. The angle is comfort guidance only. The mount, the wall, and the heat of the fireplace are the manufacturer’s instructions, not this page.
Does the rule change for ultrawide 21:9 content?
The angle stays the same. The width changes. A 21:9 or 64:27 screen is wider for the same diagonal, so the same 40° seat is farther away, and the same seat wants a smaller diagonal. Use the width. The 16:9 shortcut of about 1.1973 times the diagonal does not apply.
Does the same viewing distance apply to a projector?
Yes. A projector screen fills the same horizontal angle. Throw distance is a different question, answered from the throw ratio on the projector throw distance calculator. Seating can use the angles on this page.
Sources
- SMPTE EP 196. A recommendation of at least a 30° horizontal field of view. This page uses that as the relaxed end of the range. It is a guideline, and a living room is not a cinema.
- THX viewing-angle guidance. About 40° recommended, and about 36° for the back row. Personal preference matters.
- 20/20 visual acuity, the Snellen standard of 1 arcminute. Pixel pitch is the image height divided by the vertical pixel count. Sit at or closer than the acuity distance to resolve that format. Farther away, the extra resolution is not visible.
- Mounting. Centre the screen on seated eye height. Follow the mount and TV manufacturer instructions and the VESA pattern. This page does not say whether a wall can carry the set, and the upward angle is comfort guidance only.