Theorem of the Oilcan

In spherical Trigonometry, the theorem of the Oilcan connects the surface of a spherical Triangle to the Length on its sides; it thus constitutes a generalization of the Formule of Héron to a nonEuclidean Géométrie.

In a spherical triangle (see figure opposite) drawn on the sphere of ray R , whose sides have as angular dimensions has , B and C , one notes the half-perimeter

$p\; =\; \backslash \; frac12\; (a+b+c)\; \backslash ,$.

The theorem of the Oilcan stipulates that the surface of the triangle is worth

$S\; =\; 4R\; ^2\; \backslash \; arctan\; \backslash \; left\; \backslash \; \{\backslash \; sqrt\; \{\backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{p\}\; 2\; \backslash \; right)\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{Pa\}\; 2\; \backslash \; right)\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{Pb\}\; 2\; \backslash \; right)\; \backslash \; tan\; \backslash \; left\; (\backslash \; frac\; \{PC\}\; 2\; \backslash \; right)\}\backslash right\backslash \}$.

The formula of Héron is the borderline case of the equality above when the curve of the sphere becomes sufficiently small and that one approaches the Euclidean Géométrie: indeed, when has , B and C becomes small in front of 1 - R and large in front of BC, AC and AB - the approximation

$\backslash \; tan\; X\; \backslash \; approx\; \backslash \; arctan\; X\; \backslash \; approx\; X\; \backslash ,$

can be carried out.

See also

• Trigonometry
• Triangle
• Resolution of a triangle
• Formula of spherical Héron
• Trigonometry
• Mathematician S
• Simon the Oilcan

Random links:

The three most famous sights of Japan | Adrienne Monnier | Coronides | Hyperion university | Ryan Newman (pilot) | Banlieue_noire_de_Brownsville,_Pennsylvanie

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org