TV Size and Viewing Distance Calculator (4K, Full HD and 8K)

Choose what to start from, then enter the viewing distance or the TV size. If the "times the screen height" factors are left blank, the common guidelines are used (4K = 1.5, Full HD = 3, 8K = 0.75).

Units

Start from

Viewing distance

Aspect ratio (width:height)

Viewing distance as a multiple of the screen height (you can change it)

Full HD

Also show the viewing angle (how much of your view the screen width fills)

To check a particular TV (optional)

Size of your current or planned TV (inches, optional)

The factors and results are only guidelines. If the maker gives a recommended viewing distance for a product, go with that first.

Result and figure

Enter the viewing distance (or the TV size in inches) in the fields on the left and press "Calculate". The suggested size, viewing distance, screen dimensions, viewing angle and a drawing of the eye and the screen will appear here.

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What you can do on this page
What is this calculation used for?
How to Use
Formulas and figures
Symbols and terms
- Symbols
- Terms
Good to know before you start
How to calculate it in Excel
How to calculate it in Google Sheets
How to calculate it in Python
How to write it in LaTeX and other math languages (copy and paste)
How to have ChatGPT do the calculation
DataChef Features
Related Features
NumberChef Calculators List

What you can do on this page

Enter the distance from your seat to the TV (the viewing distance), and you get the suggested size in inches for a 4K TV and for a Full HD TV
Enter the TV size in inches, and you get the screen width, height and diagonal, plus the suggested viewing distance for 4K, Full HD and 8K
From any one of the size, width or height, you get the other dimensions (for example, "what is the largest TV that fits a 48 in wide TV stand?")
You can also check how many screen heights away you sit from your current or planned TV, and how wide an angle the screen fills in your view (the viewing angle), to compare with the SMPTE (about 30°) and THX (about 40°) guidelines
The aspect ratio (16:9, 4:3, 21:9) and the "times the screen height" factors can be changed, so it works for computer monitors and projector screens too

"About 1.5 times the screen height for 4K, 3 times for Full HD and 0.75 times for 8K" is a common guideline derived from how fine the picture is (the number of pixels). The distance and size that feel right depend on your room, what you watch and your taste. Use this calculator to narrow down your choices, not as the one size you must buy.

What is this calculation used for?

Narrowing down the right TV size for your room (buying a TV)

If it is 7 ft from the sofa to the screen, the Full HD guide is about 57 inches and the 4K guide is about 114 inches, far apart. This means "with 4K, you could go past 100 inches before the grain shows", so in practice you choose a size somewhere in between.
At the same 7 ft, a 55-inch TV fills about 32° of your view (close to the SMPTE 30°), a 65-inch about 37°, and a 75-inch about 43° (past the THX 40°). With numbers in hand, you are less likely to end up with "smaller than I expected" or "so big it tires my eyes", and you have a yardstick when you look at TVs in the store. These are only guidelines, so also check the maker's recommended distance and how the picture actually looks.

Finding the largest TV that fits a stand or a wall niche (moving or redecorating)

On a 48 in wide TV stand, the screen can be at most 48 in wide, so for 16:9 the upper limit is about \(48 \div 0.8716 \approx 55\) inches. The actual TV is larger because of the frame and stand, so in practice a 50-inch TV (about 43.6 in wide) is a likely choice.
Use "Size, width or height → screen dimensions" in this calculator to work backward from the width. For a wall mount or a built-in cabinet, check the maker's overall dimensions (width and height including the frame and stand) at the end.

Choosing a TV for a bedroom or a kid's room, watched from close up

In a bedroom 5 ft from the bed, the Full HD guide is about 41 inches and the 4K guide about 82 inches. Up close, the grain is easier to see, so this helps you decide between a small 4K TV and a Full HD TV sized for the distance.
Enter a candidate size in "Size of your current or planned TV" to see how many screen heights away you sit, and compare it with the guidelines (1.5 and 3).

Choosing a monitor size and desk depth (working from home and gaming)

A 27-inch (16:9) monitor is about 13.2 in high and 23.5 in wide. Used 24 in from your eyes, the viewing distance is about 1.8 screen heights and the viewing angle about 52°. That is close to the 4K guide (1.5 times, about 20 in), a distance that makes good use of fine text on a 4K monitor.
Ultrawide monitors (21:9) can be calculated by choosing that aspect ratio. For reading work, too wide a viewing angle means more eye and neck movement, so this is a guide for matching the desk depth to the monitor width.

Choosing a projector screen size and where to sit (home theater)

A 100-inch (16:9) screen is about 49.0 in high and 87.2 in wide. For a 4K projector the suggested distance is about 6.1 ft, and for Full HD about 12.3 ft. From 10 ft, the viewing angle is about 40°, right at the THX guideline.
The screen and the sofa are hard to move later, so measure the room first, then decide the screen size and where to sit. That way you are less likely to regret it after installing.

Checking whether the back row can see a display (offices and schools)

Seen from the back row 20 ft away, a 65-inch display (about 31.9 in high) is about 7.5 screen heights away and fills about 13.5° of the view. That is far beyond the Full HD guide (3 times), so you can expect small text on slides to be hard to read from the back.
The numbers help decide whether to add a second display, make the text larger, or move the seats forward.

Everyday uses of tan, arctan and the Pythagorean theorem (middle and high school math)

The width, height and diagonal of a screen are the three sides of a right triangle, so the Pythagorean theorem gives "the diagonal of 16:9 is \(\sqrt{16^{2}+9^{2}} = \sqrt{337}\)". The viewing angle formula \(\theta = 2\arctan(W \div 2L)\) turns the \(\tan\) of the right triangle formed by your eyes and the edge of the screen back into an angle with \(\arctan\).
It is a familiar example of how textbook formulas connect to the math of buying a TV.

Formulas and figures

Screen width, height and diagonal (from the size in inches)

Figure

Standard notation (the usual math form)

\(D\) \(=\) \(S\)

\(H\) \(=\) \(D\) \(\times\) \(\dfrac{b}{\sqrt{a^{2}+b^{2}}}\)

\(W\) \(=\) \(D\) \(\times\) \(\dfrac{a}{\sqrt{a^{2}+b^{2}}}\)

In words (symbols replaced with words)

② \(D\): diagonal (in) \(=\) ① \(S\): screen size (inches)

④ \(H\): screen height \(=\) \(D\): diagonal \(\times\) ③ height ratio (about 0.4903 for 16:9)

⑥ \(W\): screen width \(=\) \(D\): diagonal \(\times\) ⑤ width ratio (about 0.8716 for 16:9)

The formula in words

① The \(S\): screen size (inches)

② is the \(D\): diagonal (in) itself

③ Multiply the diagonal by the height ratio (\(b \div \sqrt{a^{2}+b^{2}}\) for an aspect ratio of \(a:b\))

④ to get the \(H\): screen height

⑤ Multiply the diagonal by the width ratio (\(a \div \sqrt{a^{2}+b^{2}}\))

⑥ to get the \(W\): screen width

Quick example

The diagonal, height and width of a 55-inch (16:9) TV are

\(D\): diagonal \(=\) screen size (55 in)

\(H\): screen height \(=\) diagonal (55 in) \(\times\) height ratio (0.4903)

\(W\): screen width \(=\) diagonal (55 in) \(\times\) width ratio (0.8716)

\(D = 55\ (\mathrm{in})\)

\(H = 55 \times \dfrac{9}{\sqrt{337}} \approx 26.96\ (\mathrm{in})\)

\(W = 55 \times \dfrac{16}{\sqrt{337}} \approx 47.94\ (\mathrm{in})\)

Key idea

A "55-inch" TV has a screen diagonal of 55 inches (in centimeters, 55 × 2.54 = 139.7 cm, since 1 in = 2.54 cm). But when you place a TV, what you need is the width and height, not the diagonal. When the aspect ratio (width : height) of the screen is \(a:b\), the width, height and diagonal have the same shape as a right triangle with legs \(a\) and \(b\) and hypotenuse \(\sqrt{a^{2}+b^{2}}\) (the Pythagorean theorem). So the height is \(b \div \sqrt{a^{2}+b^{2}}\) times the diagonal, and the width is \(a \div \sqrt{a^{2}+b^{2}}\) times the diagonal. For 16:9, \(\sqrt{16^{2}+9^{2}} = \sqrt{337} \approx 18.36\), so the height ratio is \(9 \div 18.36 \approx 0.4903\) and the width ratio is \(16 \div 18.36 \approx 0.8716\). In other words, for a 16:9 TV, "the height is about half the diagonal, and the width is about 0.87 times the diagonal", a handy rule for mental math in the store. These are the dimensions of the picture area (the viewable area). The actual TV is a bit larger because of the frame (bezel) and the stand, so check the maker's overall dimensions when fitting it on a TV stand or into a wall niche.

Suggested viewing distance and suggested screen size (in screen heights)

Figure

Standard notation (the usual math form)

\(L\) \(=\) \(k\) \(\times\) \(H\)

\(H\) \(=\) \(L\) \(\div\) \(k\)

\(S\) \(=\) \(H\) \(\div\) \(\dfrac{b}{\sqrt{a^{2}+b^{2}}}\)

In words (symbols replaced with words)

③ \(L\): suggested viewing distance \(=\) ② \(k\): factor (1.5 for 4K, 3 for Full HD) \(\times\) ① \(H\): screen height

⑤ \(H\): suggested screen height \(=\) ④ \(L\): viewing distance \(\div\) \(k\): factor

⑦ \(S\): suggested size (inches) \(=\) \(H\): screen height \(\div\) ⑥ height ratio (about 0.4903)

The formula in words

① Take the \(H\): screen height

② multiply it by the \(k\): factor (1.5 for 4K, 3 for Full HD, 0.75 for 8K)

③ to get the \(L\): suggested viewing distance (when the TV size is set)

④ Going the other way, divide the \(L\): viewing distance by the factor \(k\)

⑤ to get the \(H\): suggested screen height (when the viewing distance in your room is set)

⑥ Divide that height by the height ratio

⑦ to get the \(S\): suggested size (inches)

Quick example

The suggested viewing distance for a 55-inch 4K TV (screen height 26.96 in, factor 1.5), and the suggested sizes for Full HD (factor 3) and then 4K (factor 1.5) in a room with a 7 ft (84 in) viewing distance, are

\(L\): suggested viewing distance \(=\) factor (4K: 1.5) \(\times\) screen height (26.96 in)

\(H\): suggested screen height \(=\) viewing distance (84 in) \(\div\) factor (Full HD: 3)

\(S\): suggested size \(=\) screen height (28 in) \(\div\) height ratio (0.4903)

\(L = 1.5 \times 26.96 \approx 40.4\ (\mathrm{in}) \approx 3.4\ (\mathrm{ft})\)

\(H = 84 \div 3 = 28\ (\mathrm{in})\)

\(S = 28 \div 0.4903 \approx 57.1\)

\(H = 84 \div 1.5 = 56\ (\mathrm{in}),\quad S = 56 \div 0.4903 \approx 114.2\)

Key idea

The guideline "how many screen heights away to sit" comes from how fine the picture is (the number of pixels). If you sit too close, you can see the dots and the grain, so the idea is to use "the closest distance at which the grain can no longer be seen". Eyes with 20/20 vision are said to be unable to tell apart details finer than about \(\dfrac{1}{60}\) of a degree (1 arcminute). Full HD has 1,080 pixels up the screen, so one pixel (height \(H \div 1080\)) looks smaller than this angle from a distance of \(H \div \left(1080 \times \tan\left(\tfrac{1}{60}^{\circ}\right)\right) \approx 3.2H\), about 3 screen heights. 4K has twice as many pixels up the screen (2,160), so the distance is half, about 1.6 times, and 8K is half again, about 0.8 times. The calculated values are about 3.2, 1.6 and 0.8, but the rounded values "Full HD = 3, 4K = 1.5, 8K = 0.75" are commonly used as guidelines, and this calculator uses them as defaults. These factors mean "you can sit this close without noticing the grain". So think of 1.5 times for 4K as "the closest, most immersive distance", and 3 times for Full HD as "a distance where you can easily take in the whole screen". At 7 ft in the example, the Full HD guide is about 57 inches and the 4K guide is about 114 inches, far apart. This means "with 4K you could go well past 100 inches", not "with 4K you should buy a 100-inch TV". Most people choose somewhere between the two guides to suit the room, the space, the budget and what they watch. If the maker gives a recommended viewing distance for a product, go with that first. For reading text on a computer monitor, or for games where your eyes follow the edges of the screen, sitting a little farther than the TV guideline (a larger factor) can be less tiring. The factors can be changed in the input fields, so adjust them to what suits you.

Viewing angle (how much of your view the screen width fills)

Figure

Standard notation (the usual math form)

\(\theta\) \(=\) \(2\) \(\arctan\) \((\) \(W\) \(\div\) \(2L\) \()\)

In words (symbols replaced with words)

③ \(\theta\): viewing angle \(=\) \(2\) \(\arctan\) \((\) ① \(W\): screen width \(\div\) ② \(2L\): twice the viewing distance \()\)

The formula in words

① Take the \(W\): screen width

② divide it by \(2L\): twice the viewing distance , take the inverse tangent \(\arctan\) (the reverse of \(\tan\)) and double the angle

③ to get the \(\theta\): viewing angle

Quick example

The viewing angle of a 55-inch TV (screen width 47.94 in) seen from 7 ft (84 in) is

\(\theta\): viewing angle \(=\) \(2\) \(\arctan\) \((\) screen width (47.94 in) \(\div\) twice the distance (168 in) \()\)

\(\theta = 2 \arctan\dfrac{47.94}{168} = 2 \arctan 0.2854 \approx 2 \times 15.9^\circ \approx 31.9^\circ\)

Key idea

The viewing angle is how wide an angle the screen spreads across in your view. As in the figure, it is the angle between the two lines from your eyes to the left and right edges of the screen. From the center of the screen to each edge is half the width, \(W \div 2\), so in the right triangle formed by your eyes, the screen center and one edge, \(\tan(\theta \div 2) = (W \div 2) \div L = W \div 2L\). Turn this into an angle with \(\arctan\) and double it to get the viewing angle. For a movie-theater feel, guidelines from film and video industry groups are often quoted. The Society of Motion Picture and Television Engineers (SMPTE) suggests about 30°, and THX, known for certifying theater sound and picture, suggests about 36° to 40°, with 40° for the most immersive experience. On a 16:9 screen, 30° is a viewing distance of about 3.3 screen heights and 40° is about 2.4, so the Full HD guide (3 times, about 33°) is close to these. The 55-inch TV at 7 ft in the example gives about 32°, right around the SMPTE guideline. At the 4K guide (1.5 times), the angle is about 61°, like sitting near the front of a theater. Because the angle grows as you get closer, you can get the same impact with a bigger TV or by moving the sofa closer. Before buying, compare the viewing angle of your current TV with that of the TV you are considering, and you can picture in numbers how much more immersive it will be.

A TV's size in inches is its diagonal, and the Pythagorean theorem splits the diagonal by the aspect ratio (for 16:9, the height is about 0.49 times the diagonal and the width about 0.87 times). The suggested viewing distance is "screen height \(\times\) factor" (4K = 1.5, Full HD = 3, 8K = 0.75), and from a viewing distance, "height \(=\) distance \(\div\) factor" gives the suggested size. The viewing angle is \(\theta = 2\arctan(W \div 2L)\). It is about 33° at the Full HD guide (3 times), near the SMPTE 30°, and about 61° at the 4K guide (1.5 times).

Symbols and terms

Symbols

\(S\)	S	The screen size in inches, from "size". The "55" in a 55-inch TV, which is the length of the screen diagonal in inches (1 in = 2.54 cm).
\(D\)	D	The length of the screen diagonal, from "diagonal". In inches it equals \(S\); in centimeters it is \(D = 2.54S\).
\(H\)	H	The screen height, from "height". The viewing distance guideline is given as a multiple of this height.
\(W\)	W	The screen width, from "width". Used to check whether the TV fits on a stand and to calculate the viewing angle.
\(a:b\)	a to b	The aspect ratio (width : height). Current TVs and monitors are \(16:9\), old TVs and some tablets \(4:3\), and ultrawide monitors \(21:9\).
\(L\)	L	The viewing distance (in or ft), from "length". The distance from your eyes to the screen.
\(k\)	k	The factor for "viewing distance in screen heights". 4K = 1.5, Full HD = 3 and 8K = 0.75 are the common guidelines, and you can change them in this calculator.
\(\theta\)	theta	The viewing angle (degrees): the angle the screen width fills in your view. The Greek letter theta is often used for angles. Found with \(\theta = 2\arctan(W \div 2L)\).
\(\tan\)	tangent	The tangent: "opposite ÷ adjacent" in a right triangle. For half the viewing angle, \(\theta \div 2\), \(\tan(\theta \div 2) = W \div 2L\).
\(\arctan\)	arctangent inverse tangent (\(\tan^{-1}\))	The inverse tangent: returns the angle whose \(\tan\) is a given value. Also written \(\tan^{-1}\). Used to find the viewing angle from the screen width and the viewing distance.
\(\sqrt{\ }\)	square root	The square root: the positive number that gives the value when squared. Used with the Pythagorean theorem to relate the diagonal to the width and height (for example, \(\sqrt{16^{2}+9^{2}} = \sqrt{337} \approx 18.36\)).
\(\approx\)	approximately equal to	The symbol for "approximately equal to". Used when a value that does not divide evenly, or an \(\arctan\) result, is rounded to a decimal.

Terms

viewing distance	The distance from the viewer's eyes to the screen. Measure between your eyes, sitting on the sofa or a chair, and the screen. Note that it is the distance from your eyes, not from the back of the TV stand.
screen size	A TV's size, the length of the screen diagonal in inches (1 in = 2.54 cm). It is measured across the viewable picture area, so a "55-inch class" TV may measure slightly less, such as 54.6 inches.
diagonal	The line (and its length) from the bottom-left corner of the screen to the top-right corner. A TV's size in inches is this length. It is related to the width and height by the Pythagorean theorem.
aspect ratio	The ratio of the screen width to its height. Current TVs and monitors are 16:9 (16 across for every 9 up), old tube TVs and some tablets are 4:3, and ultrawide monitors are 21:9. The 21:9 in this calculator may actually be about 2.37:1 (64:27) for some products, so check the maker's specifications for exact sizes.
viewable area	The part of the TV that actually shows the picture. The width, height and diagonal from this calculator are for this area; the actual TV is a bit larger because of the frame (bezel) and the stand.
pixel	One of the tiny dots that make up the screen. Full HD has 1,920 × 1,080 pixels, 4K 3,840 × 2,160 and 8K 7,680 × 4,320. The more pixels, the less grainy the picture looks up close.
Full HD	A resolution of 1,920 pixels across and 1,080 down, also called 1080p or 2K. The suggested viewing distance is about 3 screen heights.
4K	A resolution of about 4,000 (3,840) pixels across and 2,160 down, also called UHD. It is twice Full HD in each direction (four times the pixels). The grain is harder to see, so the suggested distance is half that of Full HD (about 1.5 screen heights).
8K	A resolution of 7,680 pixels across and 4,320 down, twice 4K in each direction (four times the pixels of 4K and 16 times Full HD). The suggested distance is half that of 4K again (about 0.75 screen heights).
visual acuity	How fine a detail the eye can tell apart. Eyes with 20/20 vision are said to be unable to see details finer than about 1/60 of a degree (1 arcminute). The distance at which one pixel looks smaller than this angle is where the grain disappears, and the factors Full HD = about 3, 4K = about 1.5 and 8K = about 0.75 screen heights come from it (the calculated values are about 3.2, 1.6 and 0.8, rounded to convenient guidelines).
viewing angle	The angle the screen width fills in your view, between the two lines from your eyes to the left and right edges of the screen. It gets larger as you move closer to the same TV, and the larger it is, the more the picture fills your view.
SMPTE	The Society of Motion Picture and Television Engineers, a US-based group that sets standards for film and video technology. About 30° is often quoted as its guideline for the viewing angle from a theater seat.
THX	A brand known for certifying the sound and picture of movie theaters and home theaters. About 36° to 40° is often quoted as its guideline for the viewing angle, with 40° for the most immersive seats. Neither is a strict rule; both are guides to how immersive the picture feels.
tangent	The trigonometric ratio \(\tan\): "opposite ÷ adjacent" in a right triangle. For half the viewing angle, \(\tan(\theta \div 2) = W \div 2L\).
inverse tangent	The reverse of \(\tan\) (\(\arctan\)): it answers "what angle has this \(\tan\) value?". It is the \(\tan^{-1}\) key on a scientific calculator and the ATAN function in Excel.
radian	A unit of angle: \(180^\circ = \pi\) radians (1 radian \(\approx 57.3^\circ\)). Excel's ATAN and Python's math.atan return angles in radians, so use DEGREES (Excel) or math.degrees (Python) to read the viewing angle in degrees.
Pythagorean theorem	The rule that in a right triangle, "leg² + leg² = hypotenuse²". The width, height and diagonal of a screen are the three sides of a right triangle, so they are related by this theorem.

Good to know before you start

Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.

Converting units of length (Grades 4–5)	Knowing that 1 ft = 12 in, so you can put the viewing distance in feet and the screen dimensions in inches into the same unit Treating 1 in = 2.54 cm as "the same length in a different unit"
Ratios (Grade 6)	Understanding a ratio such as 16:9, and turning it into a multiplier (a decimal), as in "the height is 0.49 times the diagonal" Finding "how many screen heights away" with distance ÷ height
Rounding (Grades 3–4)	Reading a result such as 114.2 inches as "about 114 inches" or "around 115 inches in real products"
Right triangles and the Pythagorean theorem (Grade 8)	Knowing that "leg² + leg² = hypotenuse²" in a right triangle (for example, 3, 4, 5 and 16, 9, √337) Picturing the screen width, height and diagonal as the three sides of a right triangle
The tangent ratio and inverse trig functions (Geometry, Precalculus)	Knowing that \(\tan\theta\) is "opposite ÷ adjacent" in a right triangle, and applying it to the triangle formed by your eyes and the edge of the screen Knowing that \(\arctan\) (\(\tan^{-1}\)) returns the angle from a \(\tan\) value (a calculator or Excel is fine for the arithmetic)

How to calculate it in Excel

Copy the whole table below and paste it into cell A1 in Excel. It works as is.

Table to find the screen width, height and diagonal

Screen size S (inches)	55
Aspect ratio width a	16
Aspect ratio height b	9
Diagonal D (in)	=B1
Screen height H (in)	=B4*B3/SQRT(B2^2+B3^2)
Screen width W (in)	=B4*B2/SQRT(B2^2+B3^2)

Table to find the suggested viewing distance (screen height × factor)

Screen height H (in)	26.96
Factor k (4K=1.5, Full HD=3, 8K=0.75)	1.5
Suggested viewing distance L (in)	=B1*B2

Table to find the suggested screen size from the viewing distance

Viewing distance L (in)	84
Factor k (4K=1.5, Full HD=3, 8K=0.75)	3
Aspect ratio width a	16
Aspect ratio height b	9
Suggested screen height H (in)	=B1/B2
Suggested screen size S (inches)	=B5*SQRT(B3^2+B4^2)/B4

Table to find the viewing angle

Screen width W (in)	47.94
Viewing distance L (in)	84
Viewing angle θ (°)	=DEGREES(2ATAN(B1/(2B2)))

After pasting, the upper rows in column B are your inputs and the last rows are calculated automatically.
The first table is 55 inches at 16:9: B4 shows 55, B5 about 26.96 and B6 about 47.94. The second table is a 4K TV with a 26.96 in screen height (factor 1.5), and B3 shows about 40.4 in (about 3.4 ft). The third table is a 7 ft (84 in) viewing distance with Full HD (factor 3): B5 shows 28 and B6 about 57.1 (inches). B3 in the fourth table shows about 31.9 (°).
"SQRT" is the square root and "^2" squares a number. "ATAN" gives the inverse tangent (arctan), and because it returns radians, "DEGREES" converts the answer to degrees. To use a different factor for Full HD, 4K or 8K, change the factor cell to 1.5, 3, 0.75 and so on. For centimeters, multiply the inches by 2.54.

How to calculate it in Google Sheets

Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.

Table to find the screen width, height and diagonal

Screen size S (inches)	55
Aspect ratio width a	16
Aspect ratio height b	9
Diagonal D (in)	=B1
Screen height H (in)	=B4*B3/SQRT(B2^2+B3^2)
Screen width W (in)	=B4*B2/SQRT(B2^2+B3^2)

Table to find the suggested viewing distance (screen height × factor)

Screen height H (in)	26.96
Factor k (4K=1.5, Full HD=3, 8K=0.75)	1.5
Suggested viewing distance L (in)	=B1*B2

Table to find the suggested screen size from the viewing distance

Viewing distance L (in)	84
Factor k (4K=1.5, Full HD=3, 8K=0.75)	3
Aspect ratio width a	16
Aspect ratio height b	9
Suggested screen height H (in)	=B1/B2
Suggested screen size S (inches)	=B5*SQRT(B3^2+B4^2)/B4

Table to find the viewing angle

Screen width W (in)	47.94
Viewing distance L (in)	84
Viewing angle θ (°)	=DEGREES(2ATAN(B1/(2B2)))

The same formulas as in Excel (including SQRT, ATAN and DEGREES) work as is. Copy the whole table, paste it into cell A1, and replace the numbers in column B with your own.

How to calculate it in Python

import math

inch = 55            # screen size S (inches)
aspect_w, aspect_h = 16, 9   # aspect ratio a:b (16:9)
distance_in = 84     # viewing distance L (in); 7 ft = 84 in

# screen dimensions (in). The size in inches is the diagonal itself
diagonal_in = inch
diag_ratio = math.sqrt(aspect_w ** 2 + aspect_h ** 2)   # √(a² + b²)
height_in = diagonal_in * aspect_h / diag_ratio
width_in = diagonal_in * aspect_w / diag_ratio
print(f"Diagonal: {diagonal_in:.1f} in, height: {height_in:.2f} in, width: {width_in:.2f} in")

# suggested viewing distance (screen height × factor)
for name, k in (("4K", 1.5), ("Full HD", 3), ("8K", 0.75)):
    print(f"Suggested viewing distance for {name}: {height_in * k / 12:.2f} ft")

# suggested screen size (inches) from the viewing distance
for name, k in (("4K", 1.5), ("Full HD", 3)):
    best_height_in = distance_in / k
    best_inch = best_height_in * diag_ratio / aspect_h
    print(f"Suggested {name} size at {distance_in} in: {best_inch:.1f} in")

# viewing angle (how much of your view the screen width fills)
viewing_angle = math.degrees(2 * math.atan(width_in / (2 * distance_in)))
print(f"Viewing angle of a {inch} in TV at {distance_in} in: {viewing_angle:.1f}°")

Runs with the standard library only. math.sqrt() is the square root and math.atan() the inverse tangent (arctan); it returns radians, so math.degrees() converts to degrees. Replace the size, aspect ratio and viewing distance at the top with your own numbers and run it.

How to write it in LaTeX and other math languages (copy and paste)

Screen width, height and diagonal (from the size in inches)

D = S,  H = D × b ÷ √(a² + b²),  W = D × a ÷ √(a² + b²)

D = S,\quad H = D \cdot \frac{b}{\sqrt{a^{2}+b^{2}}},\quad W = D \cdot \frac{a}{\sqrt{a^{2}+b^{2}}}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>D</mi><mo>=</mo><mi>S</mi>
    <mo>,</mo>
    <mi>H</mi><mo>=</mo><mi>D</mi><mo>&#x22C5;</mo>
    <mfrac><mi>b</mi><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></mfrac>
    <mo>,</mo>
    <mi>W</mi><mo>=</mo><mi>D</mi><mo>&#x22C5;</mo>
    <mfrac><mi>a</mi><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></mfrac>
  </mrow>
</math>

D = S,  H = D b / sqrt(a^2 + b^2),  W = D a / sqrt(a^2 + b^2)

diag = S; {diag, diag*b/Sqrt[a^2 + b^2], diag*a/Sqrt[a^2 + b^2]}  (* D is reserved for derivatives, so the diagonal is written diag *)

diag := S;  H := diag*b/sqrt(a^2 + b^2);  W := diag*a/sqrt(a^2 + b^2);  # D is the differential operator, so the diagonal is written diag

D = S; H = D*b/sqrt(a^2 + b^2); W = D*a/sqrt(a^2 + b^2);  % S in inches, a:b is the aspect ratio (a=16, b=9 for 16:9)

D = S, H = Db/√(a^2 + b^2), W = Da/√(a^2 + b^2)

Suggested viewing distance and suggested screen size (in screen heights)

L = k × H,  H = L ÷ k,  S = H ÷ (b ÷ √(a² + b²))

L = kH,\quad H = \frac{L}{k},\quad S = \frac{H\sqrt{a^{2}+b^{2}}}{b}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>L</mi><mo>=</mo><mi>k</mi><mo>&#x2062;</mo><mi>H</mi>
    <mo>,</mo>
    <mi>H</mi><mo>=</mo><mfrac><mi>L</mi><mi>k</mi></mfrac>
    <mo>,</mo>
    <mi>S</mi><mo>=</mo>
    <mfrac>
      <mrow><mi>H</mi><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></mrow>
      <mi>b</mi>
    </mfrac>
  </mrow>
</math>

L = k H,  H = L / k,  S = H sqrt(a^2 + b^2) / b

{k*H, L/k, H*Sqrt[a^2 + b^2]/b}

L := k*H;  H := L/k;  S := H*sqrt(a^2 + b^2)/b;

L = k*H; H = L/k; S = H*sqrt(a^2 + b^2)/b;  % k is the factor (4K=1.5, Full HD=3, 8K=0.75)

L = kH, H = L/k, S = H√(a^2 + b^2)/b

Viewing angle (how much of your view the screen width fills)

θ = 2 × arctan(W ÷ (2L))

\theta = 2\arctan\frac{W}{2L}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
  <mrow>
    <mi>θ</mi><mo>=</mo><mn>2</mn><mo>&#x2062;</mo>
    <mi>arctan</mi><mo>&#x2061;</mo>
    <mfrac><mi>W</mi><mrow><mn>2</mn><mo>&#x2062;</mo><mi>L</mi></mrow></mfrac>
  </mrow>
</math>

theta = 2 arctan(W / (2 L))

2*ArcTan[W/(2*L)]*180/Pi

theta := 2*arctan(W/(2*L))*180/Pi;

theta = 2*atand(W/(2*L));  % W is the screen width, L the viewing distance (in the same unit)

θ = 2 tan^(-1)(W/(2L))

How to have ChatGPT do the calculation

You are a TV viewing distance calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing).

I will watch a 55-inch TV (aspect ratio 16:9) from a viewing distance of 7 ft (84 in).
Find each of the following:
1. The screen diagonal, height and width (in; diagonal = size in inches, height = diagonal × 9 ÷ √(16² + 9²), width = diagonal × 16 ÷ √(16² + 9²))
2. The suggested viewing distance for 4K, Full HD and 8K (screen height × 1.5, × 3 and × 0.75), in feet
3. The suggested 4K and Full HD screen sizes at 84 in (height = distance ÷ factor → diagonal → inches)
4. The viewing angle (degrees; θ = 2 × arctan(width ÷ (2 × distance)), converting radians to degrees), and how it compares with the SMPTE guideline of about 30° and the THX guideline of about 40°

Show the formulas you used and the numbers from the execution result.

How to Use

1

Enter your numbers

Type the numbers you want to calculate with into the input fields
2

Calculate

Press the "Calculate" button
3

Check the result

The result appears on the spot. The same page also explains the idea behind the calculation and the formula