Ultimate fate of the universe cosmological constant, which would correspond to a negative energy density and positive pressure, would cause even an open universe to re-collapse to a big crunch. This option has been ruled out by observations. If the average density of the universe exactly equals the critical density so that formula_8, then the geometry of the universe is flat: as in Euclidean geometry, the sum of the angles of a triangle is 180 degrees and parallel lines continuously maintain the same distance. Measurements from Wilkinson Microwave Anisotropy Probe have confirmed the universe is flat with only a 0.4% margin of error. In absence
Hyperbolic triangle side with the corresponding letter) is strictly less than a straight angle. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is called the defect of the triangle. The area of a hyperbolic triangle is equal to its defect multiplied by the square of : This theorem, first proven by Johann Heinrich Lambert, is related to Girard's theorem in spherical geometry. In all the formulas stated below the sides , , and must be measured in absolute length, a unit so that the Gaussian curvature of the plane is −1.