SSS Theorem

Specifying three sides uniquely determines a Triangle whose Area is given by Heron's Formula,

$$A = \sqrt{s(s-a)(s-b)(s-c)},$$ (1)

where

$$s\equiv {\textstyle{1\over 2}}(a+b+c)$$ (2)

is the Semiperimeter of the Triangle. Let $R$ be the Circumradius, then

$$A={abc\over 4R}.$$ (3)

Using the Law of Cosines

$$a^2 = b^2+c^2-2bc\cos A$$ (4)

$$b^2 = a^2+c^2-2ac\cos B$$ (5)

$$c^2 = a^2+b^2-2ab\cos C$$ (6)

gives the three Angles as

$$A = \cos^{-1}\left({b^2+c^2-a^2\over 2bc}\right)$$ (7)

$$B = \cos^{-1}\left({a^2+c^2-b^2\over 2ac}\right)$$ (8)

$$C = \cos^{-1}\left({a^2+b^2-c^2\over 2ab}\right).$$ (9)

See also AAA Theorem, AAS Theorem, ASA Theorem, ASS Theorem, Heron's Formula, SAS Theorem, Semiperimeter, Triangle