Geometric Proof that the Altitude of Polaris equals the Observer’s Latitude
Consider the diagram below. The circle represents the cross section of the Earth through the observer and
the dot represents the observer. The angle  represents the observer’s latitude. The angle  represents the
altitude of Polaris (the NCP really). Other angles  and  are indicated.
To Polaris
1.
2.
To zenith


3.


Horizon Plane
4.
5.
6.
7.
Proof
The two rays shown to Polaris are not actually
exactly parallel. They do converge on Polaris, but
because Polaris is so far away, over 400 lys, the ray
from the observer actually slants in toward the ray
from the Earth’s center by 9x10-11 degrees. This is
close enough to zero degrees to call the two rays to
Polaris parallel.
Angle  = 90 - , because  and  are
complementary angles of a right triangle.
Angle  equals angle  because they are opposite
angles.
Angle  = 90 - , because  and  are
complementary angles formed by the zenith line
and the horizon plane.
Since  = 90 - , and  = , then  = 90 - 
Since  = 90 - , then  = 90 - (90 - ) = 
Thus  = , or the altitude of Polaris equals the
latitude of the observer.