Saccheri quadrilateral

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Saccheri quadrilaterals


For a Saccheri quadrilateral ABCD, the sides AD and BC (also called legs) are equal in length and perpendicular to the base AB. The top CD is called the summit or upper base and the angles at C and D are called the summit angles.

The advantage of using Saccheri quardrilaterals when considering the parallel postulate is that they place the mutually exclusive options in very clear terms:


As it turns out, when the summit angles are right angles, the existence of this quadrilateral is equivalent to the statement expounded by Euclid's fifth postulate. When they are acute, this quadrilateral leads to hyperbolic geometry, and when they are obtuse, the quadrilateral leads to elliptical geometry (provided that other modifications are made to the postulates[2]). Saccheri himself, however, thought that both the obtuse and acute cases could be shown to be contradictory. He did show this in the obtuse case, but failed to properly handle the acute case.[3]








Let ABCD be a Saccheri quadrilateral having AB as base, CA and DB the equal sides that are perpendicular to the base and CD the summit. The following properties are valid in any Saccheri quadrilateral in hyperbolic geometry.[5]

  • The summit angles (at C and D) are equal and acute.
  • The summit is longer than the base.
  • The line segment joining the midpoint of the base and the midpoint of the summit is mutually perpendicular to the base and summit.
  • The above two line segments are perpendicular to each other.
  • The line segment joining the midpoint of the base and the midpoint of the summit divides the Saccheri quadrilateral into two Lambert quadrilaterals.
  • Two Saccheri quadrilaterals with congruent bases and congruent summit angles are congruent (i.e., the remaining pairs of corresponding parts are congruent).
  • Two Saccheri quadrilaterals with congruent summits and congruent summit angles are congruent.

A formula[edit]

In the hyperbolic plane of constant curvature -1, the summit s of a Saccheri quadrilateral can be calculated from the leg l and the base b using the formula

\cosh s = \cosh b \cdot \cosh^2 l - \sinh^2 l.[6]



Hyperbolic domains 2233.png
*3322 symmetry
Hyperbolic domains ii22.png
*∞∞22 symmetry

See also[edit]


  2. ^ Coxeter 1998, pg. 11
  3. ^ Faber 1983, pg. 145
  5. ^ Faber 1983, pp. 146 - 147

