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!
Draw a picture!
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."
Math Fun Facts.