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SSS Theorem

SSSTheorem

Specifying three sides uniquely determines a triangle whose area is given by Heron's formula,

K=sqrt(s(s-a)(s-b)(s-c)),
(1)

where

s=1/2(a+b+c)
(2)

is the semiperimeter of the triangle. Let R be the circumradius, then

K=(abc)/(4R).
(3)

Using the law of cosines

a^2=b^2+c^2-2bccosA
(4)
b^2=a^2+c^2-2accosB
(5)
c^2=a^2+b^2-2abcosC
(6)

gives the three angles as

A=cos^(-1)((b^2+c^2-a^2)/(2bc))
(7)
B=cos^(-1)((a^2+c^2-b^2)/(2ac))
(8)
C=cos^(-1)((a^2+b^2-c^2)/(2ab)).
(9)

See also

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

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Cite this as:

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

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