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 gauss-bonnet 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.