## Sofa Away from Me

Ever go shopping for a new TV?  Ever buy a TV, get it home, realize it's too big, and been mad you spent that money for nothing?  Ever rearrange the living room furniture, sit down because you want your MTV, and think, "Oh no!  It's so far away from me!"?  The correct relationship between size-of-television and distance-to-couch is an unsurprisingly common, but surprisingly contentious issue in this walk of life (or sit of life, as it were).  Fortunately, some trigonometry can get you out of these dire straits and make you the sultan of (couch) springs.

Diagramming your television dimensions and the distance to your couch is straightforward.  (For help finding the dimensions of your widescreen television, please see our lesson, Viewmongus.)

But this is not enough information to determine the best place to sit.  The other constants in this equation are the angles your eyeballs can resolve.  A search online reveals that a human eye can see approximately 150° left-to-right, and 135° up-and-down.  However, just because something is inside the boundary of your field of vision doesn't mean you can watch it.  Think about it: if all you could see was SCREEN, you would probably want to back away from it.  Further investigation revealed that the macula, the portion of your retina that can focus on details, can resolve an angle of 15°.  ("Macula" sounds alarmingly bloodthirsty for a thing in your eye, but if you have heard of macular degeneration, you have heard of the macula.)  The part of the screen you are reading right now is within this macular region.  However, can you read everything on the screen without moving your eyes? Probably not.  Would it still be okay to watch a program on this screen?  Maybe.

So, we've narrowed down the vertical resolvable angle for comfortable TV watching to somewhere between 15° and 135°.  Right!  At this point, we undertook several extremely scientific and precise endeavors to find a more reasonable viewing angle:

Finally, we contacted a few friends in the optical sciences.  An informal survey of practicing optometrists and ophthalmologists revealed a comfortable vertical viewing angle of 20°, and a horizontal angle of 35°.  Now, we're ready to rock.

For a given TV size and distance-from-television, we can figure out how much of our field of vision is taken up by the television.  For example, if we sit nine feet (108 in.) away from a 60-inch TV:

A quick tangent calculation reveals that the total vertical distance we see measures 38 inches.  A similar calculation on the top-down projected view, using the horizontal viewing angle, yields a horizontal viewing distance of 68 inches, hence a viewable area of 2,584 square inches (assuming this region is a rectangle, which is obviously not entirely accurate).

The 60-inch widescreen television has a screen area of 1,508 square inches.  Therefore, when we sit nine feet away, the TV takes up 58% of our viewing area.  When we sit three feet away, though, the TV comprises over 500% of our viewing area!  So now we have a fun Goldilocks problem: what distance is just right?  It turns out, by this method,  you want to sit 82 inches away from a 60-inch TV.

It also turns out that the relationship between distance-from-TV and percent-of-your-viewing-area-taken-up-by-TV is a beautiful function!  Teachers, if you are looking for a way to apply trigonometric ratios, or contextualize rational functions, check out our new lesson, Sofa Away from Me.