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SAS Theorem

SASTheorem

Specifying two sides and the angle between them uniquely (up to geometric congruence) determines a triangle. Let c be the base length and h be the height. Then the area is

K=1/2ch=1/2acsinB.
(1)

The length of the third side is given by the law of cosines,

b^2=a^2+c^2-2accosB,
(2)

so

b=sqrt(a^2+c^2-2accosB).
(3)

Using the law of sines

a/(sinA)=b/(sinB)=c/(sinC)
(4)

then gives the two other angles as

A=sin^(-1)((asinB)/(sqrt(a^2+c^2-2accosB)))
(5)
C=sin^(-1)((csinB)/(sqrt(a^2+c^2-2accosB)).)
(6)

See also

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

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Cite this as:

Weisstein, Eric W. "SAS Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SASTheorem.html

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