Calculations at an antipodal digon, spherical lune, or biangle. The digon doesn't belong to the Euclidean, but to the spherical geometry. On a sphere with the radius r, two antipodal points are connected with two straight lines with the angle α between them. So the antipodal digon ist part of a sphere surface.
Enter radius and angle and choose the number of decimal places. Then click Calculate. Please enter angles in degrees, here you can convert angle units.
Formulas:
p = 2 * π * r
A = α * 2 * r²
pi:
π = 3.141592653589793...
Radius and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
An antipodal digon is bordered by two regular great circles. A great circle is the cross-section through the center of a sphere, i.e. a circle with the same radius as the sphere. The two corners of the digon lie exactly opposite each other on a sphere. At an angle of 180 degrees, the digon corresponds to the curved surface of a hemisphere, at 360 degrees to the entire surface of a sphere, although in these two special cases it no longer has any corners. If the two-dimensional antipodal digon is extended to the third dimension, the result is a spherical wedge.
The two edges of the antipodal digon have the same curvature as the sphere on which they lie. This is only possible if both corners are opposite each other, i.e. form a semicircle. For smaller side lengths, a digon cannot be formed, but a spherical triangle can.
The basics of spherical geometry were already known in antiquity, but it was not developed as a separate branch of geometry until Leonhard Euler in the middle of the 18th century. The antipodal digon, as the simplest representative of this type of geometry, would probably not have been recognized as a separate shape by Euclid.