TV Size and Viewing Distance worksheet

1. Screen size for a 30° view (mixed use): 73.8 in

1. Convert the distance to in

   $10\times 12$$=120\text{\hspace{0.17em}in}$

2. Find the screen that fills a 30° view from your seat: width = distance × 2 tan 15°, then turn that width into a 16:9 diagonal

   $120\times 0.535898\times \frac{\sqrt{337}}{16}$$=73.7834\text{\hspace{0.17em}in}$

3. Repeat for a wider, cinema-style 40° view, using 2 tan 20° instead

   $120\times 0.72794\times \frac{\sqrt{337}}{16}$$=100.22408\text{\hspace{0.17em}in}$

4. Smallest 4K screen whose full detail you can see: 2160 pixel rows each spanning one arcminute (height ÷ 9 × √337 for the diagonal)

   $120\times 0.00029089\times \mathrm{2,160}\times \frac{\sqrt{337}}{9}$$=153.791937\text{\hspace{0.17em}in}$

2. Screen size for a 30° view (mixed use): 51.6 in

1. Convert the distance to in

   $7\times 12$$=84\text{\hspace{0.17em}in}$

2. Find the screen that fills a 30° view from your seat: width = distance × 2 tan 15°, then turn that width into a 16:9 diagonal

   $84\times 0.535898\times \frac{\sqrt{337}}{16}$$=51.64838\text{\hspace{0.17em}in}$

3. Repeat for a wider, cinema-style 40° view, using 2 tan 20° instead

   $84\times 0.72794\times \frac{\sqrt{337}}{16}$$=70.156856\text{\hspace{0.17em}in}$

4. Smallest 8K screen whose full detail you can see: 4320 pixel rows each spanning one arcminute (height ÷ 9 × √337 for the diagonal)

   $84\times 0.00029089\times \mathrm{4,320}\times \frac{\sqrt{337}}{9}$$=215.308712\text{\hspace{0.17em}in}$

3. Distance for a 30° view (mixed use): 8.1 ft

1. Turn the advertised diagonal into the screen's width, because the viewing angle depends on width

   $60\times \frac{16}{\sqrt{337}}$$=52.294532\text{\hspace{0.17em}in}$

2. 30° view: distance = width ÷ (2 tan 15°), converted to ft

   $\frac{52.294532}{0.535898\times 12}$$=8.13191\text{\hspace{0.17em}ft}$

3. 40° view: distance = width ÷ (2 tan 20°), converted to ft

   $\frac{52.294532}{0.72794\times 12}$$=5.986585\text{\hspace{0.17em}ft}$

4. 4K detail: height = diagonal × 9 ÷ √337 = 29.42 in; one pixel row (height ÷ 2160) spans one arcminute at this distance

   $\frac{29.415674}{\mathrm{2,160}\times 0.00029089\times 12}$$=3.901375\text{\hspace{0.17em}ft}$

4. Distance for a 30° view (mixed use): 9.5 ft

1. Turn the advertised diagonal into the screen's width, because the viewing angle depends on width

   $70\times \frac{16}{\sqrt{337}}$$=61.010288\text{\hspace{0.17em}in}$

2. 30° view: distance = width ÷ (2 tan 15°), converted to ft

   $\frac{61.010288}{0.535898\times 12}$$=9.487229\text{\hspace{0.17em}ft}$

3. 40° view: distance = width ÷ (2 tan 20°), converted to ft

   $\frac{61.010288}{0.72794\times 12}$$=6.984349\text{\hspace{0.17em}ft}$

4. 4K detail: height = diagonal × 9 ÷ √337 = 34.32 in; one pixel row (height ÷ 2160) spans one arcminute at this distance

   $\frac{34.318287}{\mathrm{2,160}\times 0.00029089\times 12}$$=4.551604\text{\hspace{0.17em}ft}$

5. Screen size for a 30° view (mixed use): 147.6 cm

1. Convert the distance to cm

   $2.4\times 100$$=240\text{\hspace{0.17em}cm}$

2. Find the screen that fills a 30° view from your seat: width = distance × 2 tan 15°, then turn that width into a 16:9 diagonal

   $240\times 0.535898\times \frac{\sqrt{337}}{16}$$=147.566799\text{\hspace{0.17em}cm}$

3. Repeat for a wider, cinema-style 40° view, using 2 tan 20° instead

   $240\times 0.72794\times \frac{\sqrt{337}}{16}$$=200.44816\text{\hspace{0.17em}cm}$

4. Smallest 8K screen whose full detail you can see: 4320 pixel rows each spanning one arcminute (height ÷ 9 × √337 for the diagonal)

   $240\times 0.00029089\times \mathrm{4,320}\times \frac{\sqrt{337}}{9}$$=615.167749\text{\hspace{0.17em}cm}$

6. Screen size for a 30° view (mixed use): 44.3 in

1. Convert the distance to in

   $6\times 12$$=72\text{\hspace{0.17em}in}$

2. Find the screen that fills a 30° view from your seat: width = distance × 2 tan 15°, then turn that width into a 16:9 diagonal

   $72\times 0.535898\times \frac{\sqrt{337}}{16}$$=44.27004\text{\hspace{0.17em}in}$

3. Repeat for a wider, cinema-style 40° view, using 2 tan 20° instead

   $72\times 0.72794\times \frac{\sqrt{337}}{16}$$=60.134448\text{\hspace{0.17em}in}$

4. Smallest 8K screen whose full detail you can see: 4320 pixel rows each spanning one arcminute (height ÷ 9 × √337 for the diagonal)

   $72\times 0.00029089\times \mathrm{4,320}\times \frac{\sqrt{337}}{9}$$=184.550325\text{\hspace{0.17em}in}$

7. Distance for a 30° view (mixed use): 1.8 m

1. Turn the advertised diagonal into the screen's width, because the viewing angle depends on width

   $110\times \frac{16}{\sqrt{337}}$$=95.873309\text{\hspace{0.17em}cm}$

2. 30° view: distance = width ÷ (2 tan 15°), converted to m

   $\frac{95.873309}{0.535898\times 100}$$=1.78902\text{\hspace{0.17em}m}$

3. 40° view: distance = width ÷ (2 tan 20°), converted to m

   $\frac{95.873309}{0.72794\times 100}$$=1.317049\text{\hspace{0.17em}m}$

4. 4K detail: height = diagonal × 9 ÷ √337 = 53.93 cm; one pixel row (height ÷ 2160) spans one arcminute at this distance

   $\frac{53.928736}{\mathrm{2,160}\times 0.00029089\times 100}$$=0.858302\text{\hspace{0.17em}m}$

8. Screen size for a 30° view (mixed use): 59 in

1. Convert the distance to in

   $8\times 12$$=96\text{\hspace{0.17em}in}$

2. Find the screen that fills a 30° view from your seat: width = distance × 2 tan 15°, then turn that width into a 16:9 diagonal

   $96\times 0.535898\times \frac{\sqrt{337}}{16}$$=59.02672\text{\hspace{0.17em}in}$

3. Repeat for a wider, cinema-style 40° view, using 2 tan 20° instead

   $96\times 0.72794\times \frac{\sqrt{337}}{16}$$=80.179264\text{\hspace{0.17em}in}$

4. Smallest 8K screen whose full detail you can see: 4320 pixel rows each spanning one arcminute (height ÷ 9 × √337 for the diagonal)

   $96\times 0.00029089\times \mathrm{4,320}\times \frac{\sqrt{337}}{9}$$=246.0671\text{\hspace{0.17em}in}$

9. Screen size for a 30° view (mixed use): 227.5 cm

1. Convert the distance to cm

   $3.7\times 100$$=370\text{\hspace{0.17em}cm}$

2. Find the screen that fills a 30° view from your seat: width = distance × 2 tan 15°, then turn that width into a 16:9 diagonal

   $370\times 0.535898\times \frac{\sqrt{337}}{16}$$=227.498816\text{\hspace{0.17em}cm}$

3. Repeat for a wider, cinema-style 40° view, using 2 tan 20° instead

   $370\times 0.72794\times \frac{\sqrt{337}}{16}$$=309.024246\text{\hspace{0.17em}cm}$

4. Smallest 8K screen whose full detail you can see: 4320 pixel rows each spanning one arcminute (height ÷ 9 × √337 for the diagonal)

   $370\times 0.00029089\times \mathrm{4,320}\times \frac{\sqrt{337}}{9}$$=948.383613\text{\hspace{0.17em}cm}$

10. Distance for a 30° view (mixed use): 3.7 m

1. Turn the advertised diagonal into the screen's width, because the viewing angle depends on width

   $225\times \frac{16}{\sqrt{337}}$$=196.104496\text{\hspace{0.17em}cm}$

2. 30° view: distance = width ÷ (2 tan 15°), converted to m

   $\frac{196.104496}{0.535898\times 100}$$=3.65936\text{\hspace{0.17em}m}$

3. 40° view: distance = width ÷ (2 tan 20°), converted to m

   $\frac{196.104496}{0.72794\times 100}$$=2.693963\text{\hspace{0.17em}m}$

4. 1080p detail: height = diagonal × 9 ÷ √337 = 110.31 cm; one pixel row (height ÷ 1080) spans one arcminute at this distance

   $\frac{110.308779}{\mathrm{1,080}\times 0.00029089\times 100}$$=3.511237\text{\hspace{0.17em}m}$