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Circles and disks

In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r {\displaystyle 2\pi r}.

Let R = 1 − K {\displaystyle R={\frac {1}{\sqrt {-K}}}}, where K {\displaystyle K} is the Gaussian curvature of the plane. In hyperbolic geometry, K {\displaystyle K} is negative, so the square root is of a positive number.

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

And the area of the enclosed disk is:

Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than 2 π {\displaystyle 2\pi }, 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: 1 tanh (r) {\displaystyle {\frac {1}{\tanh(r)}}}