In solid geometry, the area of a spherical quadrangle can be exactly calculated. A spherical quadrangle is the intersection of a lune and a zone. In geographic terms, a quadrangle is defined as a region bounded by parallels north and south, and meridians east and west.
In the pictured example, a quadrangle is formed by the intersection of a zone, which is the region bounded by 15°N and 45°N latitudes, and a lune, which is the region bounded by 0° and 30°E longitude. Under the spherical planet assumption, the fraction of the entire spherical surface area inscribed in the quadrangle can be calculated:
area = areaquad(15,0,45,30) area = 0.0187
That is, less than 2% of the planet's surface area is in this quadrangle. To get an absolute figure in, for example, square miles, you must provide the appropriate spherical radius. The radius of the Earth is about 3958.9 miles:
area = areaquad(15,0,45,30,3958.9) area = 3.6788e+06
The surface area within this quadrangle is over 3.6 million square miles for a spherical Earth.