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SAS Theorem


Specifying two sides and the Angle between them uniquely determines a Triangle. Let $b$ be the base length and $h$ be the height. Then the Area is

K = {\textstyle{1\over 2}}ch = {\textstyle{1\over 2}}ac\sin B.
\end{displaymath} (1)

The length of the third side is given by the Law of Cosines,

b^2=a^2+c^2-2ac\cos B,

b=\sqrt{a^2+c^2-2ac\cos B}.
\end{displaymath} (2)

Using the Law of Sines
{a\over\sin A}={b\over\sin B}={c\over\sin C}
\end{displaymath} (3)

then gives the two other Angles as
$\displaystyle A$ $\textstyle =$ $\displaystyle \sin^{-1}\left({a\sin B\over\sqrt{a^2+c^2-2ac\cos B}}\right)$ (4)
$\displaystyle C$ $\textstyle =$ $\displaystyle \sin^{-1}\left({c\sin B\over\sqrt{a^2+c^2-2ac\cos B}}.\right)$ (5)

See also AAA Theorem, AAS Theorem, ASA Theorem, ASS Theorem, SSS Theorem, Triangle

© 1996-9 Eric W. Weisstein