The Most Valuable Unit

There is a common measurement that could be extremely useful to us but is rarely used. Whenever we look at an object, it manifests a certain angular diameter in our field of vision. The angular diameter of any object in our vision is proportional to it's actual diameter divided by it's distance from us. Of course, if it is an unevenly sided object, trigonometric functions must come into play.

But measuring and expressing angular diameter is something that we handle very poorly. We usually express angular diameter in imprecise, subjective terms such as "loomed large in the sky".

I believe that getting in the habit of precision measurement of angular diameter would be very useful because it gives us an immediate ratio of the diameter of the subject in relation to it's distance from us. This ratio determines how much of our field of vision, or that of a camera, the object will occupy. Angular diameter could be measured at least as easily as physical diameter using a simple scope or square.

The beginning of the problem is the way that we measure angles. We measure in degrees. 90 degrees is a right angle, 180 is a straight line and 360 is a complete circle.

Measurement in degrees works just fine if we are measuring the outside of a circle, such as latitude and longitude on the earth's surface. But it is my conclusion that we must deal with circles and spheres as two different entities if we are measuring the circle from the inside, instead of the outside.

Whenever we look at an object some distance away, we are forming a circle with us at the center and the object on the edge of the circle. If there is vertical elevation involved, then we form a three-dimensional sphere rather than a two-dimensional circle.

We do sometimes express angular diameter, such as the apparent distance between two stars in the sky. However, the reason that we are not making full use of thus measurement is that we try to measure in degrees. While this works fine for the outside of a sphere or circle, it works very poorly for measuring the inside and this is what we are doing when we look at an object some distance away.

My solution is to dig out that old standby of math class known as a radian. This is simply the distance on a circle equal to the radius of the circle, meaning that there are 2 pi radians in a complete circle. This is a textbook unit that is rarely used in the "real world" and I find that to be a shame because it could easily be extremely useful. Since measurement in radians is a ratio, the actual diameter of the object divided by it's distance from us, we can treat measurement of angular diameter in radians in the same way as trigonometric functions, which are also ratios.

If you look at an object such as a building perpendicular to it's center from a distance equal to the width of the building, it will occupy an angular diameter of one radian. We only have to keep in mind that this angular diameter is the arc of the circle of which the observer is at the center and not the straight line side of the building, which is actually a chord on the circle. This means that if we look at the side of a building along a line perpendicular to the side, we actually have to subtract the sine of the angle which is half the angular diameter of the building.

If the building occupies an angular diameter of 1.57 radians, which is 90 degrees, we would take the sine of half that angle, 45 degrees, which is 0.5. So, if we subtract 0.5 of 1.57 radians, we get 0.785. This means that the side of the rectangular building that we are looking at along a line perpendicular to it's side and which occupies an angular diameter of 1.57 radians, is at a distance from us of 0.785 the length of the side of the building.

Probably the reason that we do not make more use of this versatile unit is that it does not fit with our base ten number system. There is 2 pi, or 6.28 radians in a complete circle. The conversion factor with degrees is 57.3. This means that if you looked at an object from a distance ten times the object's diameter, it would occupy an angular diameter of .1 radian or 5.73 degrees.

The radian would appear to be especially useful in space travel because when a spherical body, such as a planet, occupies an angular diameter of one radian, it means that the observer is at a distance from the planet's surface equal to the radius of the planet, or the distance from it's surface to it's center.

One reason that the radian is such an interesting unit is that it is actually the only completely natural unit that human beings use. The vast majority of units that we use were created arbitrarily. Seconds, meters, feet, miles and, pounds are entirely arbitrary. A degree was chosen to be 1/360 of a complete circle because 360 is a very round and easily divisible number, but even so this is an arbitrary unit that is meaningless in the universe of inanimate matter.

Days and years can be said to be natural units because they are based on the rotation and revolution of the earth. Yet these are natural units only on earth or while measuring it. Further out in space, the day and the year are just as arbitrary as the other units.

Radians, however, are a natural unit anywhere in the universe and as far as I can see, is the only unit of which this claim can be made.