Chapter 8.P, Problem 3P

If a sphere of radius r is sliced by a plane whose distance from the centre of the sphere is d, then the sphere is divided into two pieces called segments of one base (see the first figure).

The corresponding surfaces are called spherical zones of one base.

(a) Determine the surface areas of the two spherical zones indicated in the figure.

(b) Determine the approximate area of the Arctic Ocean by assuming that it is approximately circular in shape, with center at the North Pole and “circumference” at $75°$ north latitude. Use r = 3960 mi for the radius of the earth.

(c) A sphere of radius r is inscribed in the right circular cylinder of radius r. Two planes perpendicular to the central axis of the cylinder and a distance h apart cut off a spherical zone of two bases on the sphere (see the second figure). Show that the surface area of the spherical zone equals the surface area of the region that the two planes cut off on the cylinder.

(d) The Torrid Zone is the region on the surface of the earth that is between the Tropic of Cancer ( $23.45°$ north latitude) and the Tropic of Capricorn ( $23.45°$ south latitude).What is the area of the Torrid Zone?