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?

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- Dec 2nd 2010, 12:51 PMchriscnon-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, 01:54 PMemakarov
According to the Girard's theorem, $\displaystyle A = R^2 \cdot E$ 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 - $\displaystyle \pi$. And, as I remember, curvature is the inverse of the radius.

- Dec 3rd 2010, 04:57 AMHallsofIvy