Calculations

(1) Small-Angle Formula: D = θ d / 206,265 arcseconds, where the number of arcseconds in a circle divided by 2π = 206,265

D = linear size of an object

θ = angular size of the object, in arcseconds

1° = 60 arcminutes = 3600 arcseconds

d = distance to the object

Angular size of a pinhead

θ_Moon = 0.5°, which is roughly the size of a large pea (8 mm or 0.314 in) held at arm's length:

(2) 0.5° = arc tangent(D / 36 inches)

D = 0.314 inches

By this equation, we get that the angular size of a pin is:

arc tangent(0.0590551 inches / 36 inches) = θ_pin

Where a pin head is 1.5 mm = 0.06 in. wide

θ_pin = 0.095° = 344 arcsec

Calculate distance for pinhead-sized Death Star

(1) D = (θ_pin * d) / 206,265 arcsec

d = D * 206,265 / θ_pin

d = 120 km * 206,265 / 344 arcsec

d = 71,953 km

Calculate size at Moon's distance from Earth

(1) D = (θ_{Death Star} x d) / 206,265 arcsec

θ_{Death Star} = D x 206,265 arcsec / d

θ_{Death Star} = 120 km x 206,265 arcsec / 385,000 km

θ_{Death Star} = 64 arcseconds ≈ 1 arcminute

Calculate distance to appear as large as the Moon

(1) D = (θ_Moon x d) / 206,265 arcsec

d = 120 km * 206,265 arcsec / 1860 arcsec

d = 13,307 km = 8,264 mi, which is roughly the distance of a direct flight from New York to Hyderabad, India.