# non-euclidean

• Dec 2nd 2010, 01:51 PM
chrisc
non-euclidean
A bug named Fred lives in a 2 dimensional universe on constant curvature. He draws a ΔABC of area 108 km2 and finds that μ (∠A) = 100°, μ (∠B )= 90° and μ(∠C)= 10°, What kind of universe does he live in? What is its curvature and radius?
• Dec 2nd 2010, 02:54 PM
emakarov
According to the Girard's theorem, $A = R^2 \cdot E$ where A is the triangle area, R is the radius of the sphere (it's the sphere because the sum of the angles is greater than 180°) and E = ∠A + ∠B + ∠C - $\pi$. And, as I remember, curvature is the inverse of the radius.
• Dec 3rd 2010, 05:57 AM
HallsofIvy
Quote:

Originally Posted by emakarov
According to the Girard's theorem, $A = R^2 \cdot E$ where A is the triangle area, R is the radius of the sphere (it's the sphere because the sum of the angles is greater than 180°) and E = ∠A + ∠B + ∠C - $\pi$.

which means that the angles must be given in radians here.

Quote:

And, as I remember, curvature is the inverse of the radius.