How would i show the following:
Show that about 4% of the Earth's surface lies north of the Arctic circle (at latitude +66.5 degrees).
or look here:Spherical Cap -- from Wolfram MathWorld
Since you are looking only for a per cent value, you do not require the radius of the earth in miles or kilometers, just use "R" for the radius.
Calculate the total surface area of a sphere,
then the area of the spherical cap.
Surface Area of a spherical cap
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.
replacing h (in the spherical cap equation above)