Originally Posted by
aidan Surface Area of a sphere:
$\displaystyle S_{\text{sphere}} = 4 \, \pi \, r^2 $
Surface Area of a spherical cap
$\displaystyle S_{\text{cap}} = 2 \, \pi \, r \, h $
If you sliced the earth along the latitude 66.5 degress,
the distance from the plane ( the flat part )
to the north point is the distance h.
$\displaystyle h = R - R \sin (latitude) $
$\displaystyle h = R(1 - \sin (latitude)) $
replacing h (in the spherical cap equation above)
$\displaystyle S_{\text{cap}} = 2 \, \pi \, r^2 (1-sin(latitude)) $
the ratio:
$\displaystyle \dfrac{ \text{SphericalCapArea}}{\text{SurfaceAreaOfSphere }}$
$\displaystyle \dfrac{2\,\pi\,r^2(1-\sin(latitude))}{4\,\pi\,r^2} $
Radius cancels.
Pi cancels.
.