Editing Hyperbolic geometry (section)

=== Circles and disks ===

In hyperbolic geometry, the circumference of a circle of radius ''r'' is greater than <math> 2 \pi r </math>.

Let <math> R =  \frac{1}{\sqrt{-K}}  </math>, where <math> K </math> is the [[Gaussian curvature]] of the plane.  In hyperbolic geometry, <math>K</math> is negative, so the square root is of a positive number.

Then the circumference of a circle of radius ''r'' is equal to:

:<math>2\pi R \sinh \frac{r}{R} \,.</math>

And the area of the enclosed disk is:

:<math>4\pi R^2 \sinh^2 \frac{r}{2R} = 2\pi R^2 \left(\cosh \frac{r}{R} - 1\right) \,.</math>

Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than <math> 2\pi </math>, though it can be made arbitrarily close by selecting a small enough circle.

If the Gaussian curvature of the plane is −1 then the [[geodesic curvature]] of a circle of radius ''r''  is: <math> \frac{1}{\tanh(r)} </math><ref name="auto">{{cite web|url=https://math.stackexchange.com/q/2430495/88985|website=math [[stackexchange]]|title= Curvature of curves on the hyperbolic plane|access-date=24 September 2017}}</ref>