In euclidean geometry, the notion of area is based on the area of a rectangle. The area of parallelograms and triangles follows easily from this definition. The area of other regions is determined by a limiting process involving an approximation by rectangles. In hyperbolic geometry, we do not have figures analogous to rectangles. A hyperbolic quadrilateral has angle sum less than 2π so cannot have four right angles. Instead, we use triangles as basic figures.
the gaussbonnet formula For the moment, we shall regard this as the definition of the hyperbolic area.


Note. If we divide ΔABC by adding a hyperbolic cevian CD with D on AB , then the formula gives the expected result that area(ABC) = area(ACD)+area(BCD). This follows from the diagram on the right.
ΔABC has angles α,β and γ = γ'+γ", so area(ABC)= π(α+β+γ).
It follows that, if we divide a hyperbolic figure into hyperbolic triangles by adding
In this way we can find the hyperbolic area of a hyperbolic polygon (i.e. a figure
the hyperbolic polygon theorem The 2 in the formula arises since tha angles round the interior point add up to 2π.
The figure on the right shows a convex hyperbolic pentagon (in red). The green
In euclidean geometry, there are many ways to find the area of a triangle.


