# 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

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:

${\displaystyle \theta _{0}={\frac {w}{r}}={\frac {111}{r}}}$

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:

${\displaystyle \theta _{1}={\frac {111}{\sqrt {r^{2}+46^{2}}}}}$

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:

${\displaystyle \theta _{2}=2\arctan {\left({\frac {111}{2{\sqrt {r^{2}+46^{2}}}}}\right)}}$

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

${\displaystyle \arctan u=u-{\tfrac {1}{3}}u^{3}+{\tfrac {1}{5}}u^{5}-{\tfrac {1}{7}}u^{7}}$

At 20 degrees, the conversion to radians yields u≈.35 and u3≈.04 so that the first non-zero correction term changes the result by only 4%.

#### results

 ${\displaystyle \theta _{0}}$-up(degrees) ${\displaystyle \theta _{0}}$-down(degrees) ${\displaystyle \theta _{1}}$-up(degrees) ${\displaystyle \theta _{1}}$-down(degrees) ${\displaystyle \theta _{2}}$-up(degrees) ${\displaystyle \theta _{2}}$-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