# Physics and Astronomy Labs/Angular size/Comparison of two hand positions for 20 degrees

## The engineers look at the two hand positions for the 20 degree measurement[edit]

Ten students measured the angular size to be 20 degrees by standing in front of an object that is 111 inches long and using one of two hand positions: "up" if the middle three fingers are up, or "down" if they are down. The rows indicate three steps in the calculation: the first two rows ("0") used the zero order approximation:

where w=111 inches was the width of the object, and r is the distance from the object (in inches also). The 111 inch long object was the top of an exit to a building that was approximately 46 inches higher that a typical eye. If r denotes the horizontal distance as measured along the floor, the actual distance is a bit larger than r:

A second correction to consider involves the fact that w is not an arclength, but a straight line segment. From the relationship between a cord and the angle swept by that cord, we finally reach our final (and best) form for angular diameter:

The connection between the aforementioned two equations is made using the well-known Taylor series expansion of the arctan function^{[1]}:

At 20 degrees, the conversion to radians yields u≈.35 and u^{3}≈.04 so that the first non-zero correction term changes the result by only 4%.

#### results[edit]

-up (degrees) |
-down (degrees) |
-up (degrees) |
-down (degrees) |
-up (degrees) |
-down (degrees) | |||

22.63 | 18.82 | 22.35 | 18.65 | 22.06 | 18.48 | |||

18.93 | 18.82 | 18.76 | 18.65 | 18.59 | 18.48 | |||

17.47 | 16.96 | 17.34 | 16.84 | 17.20 | 16.71 | |||

19.63 | 19.39 | 19.44 | 19.21 | 19.25 | 19.03 | |||

21.49 | 18.33 | 21.24 | 18.17 | 20.99 | 18.02 | |||

18.82 | 20.25 | 18.65 | 20.05 | 18.48 | 19.84 | |||

22.16 | 22.63 | 21.89 | 22.35 | 21.62 | 22.06 | |||

22.16 | 19.63 | 21.89 | 19.44 | 21.62 | 19.25 | |||

20.52 | 20.78 | 20.30 | 20.56 | 20.09 | 20.34 | |||

20.72 | 20.06 | 20.49 | 19.86 | 20.27 | 19.66 | |||

mean | 20.32 | 19.46 | 20.11 | 19.28 | 19.90 | 19.09 | ||

stdev | 1.71 | 1.53 | 1.66 | 1.49 | 1.61 | 1.44 |