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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Semiperimeter Formulas">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed Feb 12 16:13:47 1997}
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\section{Semiperimeter Formulas and the Second Law of Tangents}

In a triangle, the \textbf{semiperimeter} is one half the length of the
perimeter. This quantity arises naturally during the course of many
computations involving general triangles.

\begin{theorem}
If the lengths of the sides of a triangle are $A$, $B,$ and $C$, then the
radius $r$ of the inscribed circle is given by 
\[
r=\sqrt{\frac{\left( S-A\right) \left( S-B\right) \left( S-C\right) }{S}}
\]
\medskip 
\end{theorem}

One application of the semiperimeter is the \textbf{second law of tangents}.
The second law of tangents may be used instead of the law of cosines to
solve a triangle where three sides are given; it is somewhat better suited
to computation.

\begin{theorem}
(Second Law of Tangents\label{second law of tangents}) If $A$, $B$, and $C$
are the lengths of the sides of a triangle, $\gamma $ is the angle opposite $%
C$, and $S=\frac{1}{2}\left( A+B+C\right) $ is the semiperimeter\label%
{semiperimeter}, then 
\[
\tan \frac{\gamma }{2}=\sqrt{\frac{\left( S-A\right) \left( S-B\right) }{%
S\left( S-C\right) }}
\]
\end{theorem}

\begin{example}
Given three sides $A=5$, $B=8$, and $C=12$ , then $S=\dfrac{5+8+12}{2}%
=\allowbreak \dfrac{25}{2}$, and the angle $\gamma $ opposite $C$ is found
by solving the equation 
\[
\tan \frac{\gamma }{2}=\sqrt{\frac{\left( \dfrac{25}{2}-5\right) \left( 
\dfrac{25}{2}-8\right) }{\dfrac{25}{2}\left( \dfrac{25}{2}-12\right) }}
\]
for $\gamma $, yielding 
\[
\gamma =2\arctan \frac{3}{5}\sqrt{15}\approx \allowbreak 2.\,\allowbreak
328\,8\approx 133.\,\allowbreak 43\unit{\UNICODE{0xb0}}
\]
\end{example}

\subsection{\ Related Reference Topics}

{\small \hyperref{Algebra}{}{}{Al_contents.tex}}

{\small \hyperref{Geometry}{}{}{GE_contents.tex}}

{\small \hyperref{Calculus}{}{}{CAL_contents.tex}}

{\small \hyperref{Tables of Trigonometric Facts}{}{}{TAB_trigonometry.tex}}%
\hfill

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Trigonometry}{}{}{../../Help/DM4.tex}}

{\small \hyperref{Function Definitions}{}{}{../../Help/DM5.tex}}

{\small \hyperref{Numbers, Functions, and Units}{}{}{../../Help/DM2.tex}}

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