Spherical Rectangle Calculator

Calculations for a spherical rectangle. A spherical rectangle is formed by four great-circle arcs on a sphere. Pairs of opposite sides lie on the same great circle and are of equal length. In spherical geometry, also the great circles containing opposite sides intersect. Therefore, the sides of the spherical rectangle must lie between the intersection points of the corresponding great circles. A great circle containing side a intersects a great circle containing side b at an angle α greater than 90 degrees.
Please enter the radius of the sphere and the side lengths of the spherical rectangle. The other values ​​will be calculated.

	Sphere radius:
	Side a:
	Side b:
	Arc diagonal (d):
	Vertex angle (α):
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Radius, lengths, diagonal and perimeter have a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter). A spherical rectangle is two-dimensional but extends into three dimensions.

A spherical rectangle belongs to non-Euclidean geometry, specifically to elliptic geometry, and more precisely to spherical geometry. In such a geometry, there are no parallel lines. Like a planar rectangle, a spherical rectangle has opposite sides of equal length that do not intersect within the rectangle itself. The four angles are equal in size, though the right angles implied by the name do not exist here. The two diagonals of the spherical rectangle are of equal length and also lie along great circles. The sum of the angles of a spherical rectangle exceeds 360 degrees, just as the sum of the angles of a spherical triangle exceeds 180 degrees.
Another variant of a rectangle on a sphere is the lesser-known sphere-cut rectangle, which features angles smaller than 90 degrees.

The formulas for a spherical rectangle are derived from the properties of spherical geometry. The side lengths a and b are treated as the lengths of great-circle arcs and converted into the corresponding central angles of the sphere. Using the spherical law of cosines, the angles and the length of the diagonal can be determined from these values. The area is determined by the spherical excess, that is, the difference between the sum of the spherical rectangle's angles and 360 degrees.

Calculation of area and other dimensions: Spherical Rectangle

Last updated on 08/12/2026. Author: Jürgen Kummer

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