Area of a Hyperbolic Wedge

The unit circle can be parametrized by (cos w, sin w). Given a point on it, the region cut out by this circle, the x-axis, and the ray from the origin to this point has area w/2.

As you may know, the hyperbolic cosine and hyperbolic sine functions are similar to the usual cosine and sine; they obey similar properties. Here's a cool fact that is parallel to the one above.

The parametrization (cosh w, sinh w) parametrizes a hyperbola. Given a point on it, the region cut out by this hyperbola, the x-axis, and the ray to from the origin to this point also has area w/2!

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The Math Behind the Fact:
The above fact can be verified by integration in polar coordinates.

How to Cite this Page:
Su, Francis E., et al. "Area of a Hyperbolic Wedge." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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