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Triangles

Unlike Euclidean triangles, where the angles always add up to π radians (180°, a straight angle), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radians (180°, a straight angle). The difference is referred to as the defect. Generally, the defect of a convex hyperbolic polygon with n {\displaystyle n} sides is its angle sum subtracted from (n − 2) ⋅ 180 ∘ {\displaystyle (n-2)\cdot 180^{\circ }}.

The area of a hyperbolic triangle is given by its defect in radians multiplied by R, which is also true for all convex hyperbolic polygons. As a consequence, all hyperbolic triangles have an area that is less than or equal to R π. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.

As in Euclidean geometry, each hyperbolic triangle has an incircle. In hyperbolic geometry, if all three of its vertices lie on a horocycle or hypercycle, then the triangle has no circumscribed circle.

As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent.