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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Trapezoid">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Mon Feb 10 15:54:28 1997}
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\begin{document}


\section{Trapezoid\label{Trapezoid}}

\begin{definition}
A trapezoid is a four-sided figure with at least two sides parallel.
\end{definition}

Let $a$ and $b$ be the lengths of the two parallel sides, $c$ and $d$ the
lengths of the other sides, and $h$ the height (the distance between the two
parallel sides).\FRAME{dtbpF}{183.6875pt}{121.5625pt}{0pt}{}{}{Figure }{%
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\begin{itemize}
\item  The \label{Perimeter of trapezoid}\textbf{perimeter} of a trapezoid
is the sum of the lengths of the sides.

$P=a+b+c+d$

\item  The \label{Area of trapezoid}\textbf{area} of a trapezoid is the
height times the average of the lengths of two parallel sides.

$A=\frac{1}{2}\left( a+b\right) h$
\end{itemize}

\begin{example}
The perimeter of the trapezoid with sides of length $a=3$, $b=5$, $c=3$, and 
$d=2$ is 
\[
P=3+5+3+2=\allowbreak 13 
\]
The area of the trapezoid with parallel sides of length $a=3$, $b=5$ and of
height $h=2$ is 
\[
A=\frac{1}{2}\left( 3+5\right) 2=\allowbreak 8 
\]
\end{example}

\begin{example}
To find the area of a trapezoid given only the length of the sides, observe
that sometimes the area can be computed as the sum of the areas of a
triangle and of a rectangle. \FRAME{dtbpF}{266.9375pt}{93.625pt}{0pt}{}{}{%
Figure }{\special{language "Scientific Word";type
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sides of length $a=2$ and $b=6$ and two other sides of length $c=3$, and $%
d=2 $, then the triangle has sides of length $c=3$ and $d=2$ and base of
length $b-a=6-2=4$. The example provided earlier for triangles gets $h=\frac{%
3}{8}\sqrt{15}$ (using the Pythagorean Theorem). Thus the area of the
trapezoid is 
\[
A=\frac{1}{2}\left( 2+6\right) \frac{3}{8}\sqrt{15}=\allowbreak \frac{3}{2}%
\sqrt{15}\approx \allowbreak 5.\,\allowbreak 809 
\]
$\,$
\end{example}

\subsection{\ Related Reference Topics}

{\small \hyperref{Pythagorean Theorem}{}{}{GE00_01.tex}}

{\small \hyperref{Table of Geometric Formulas For Length, Area, and Volume}{%
}{}{TAB_geom_formulas.tex}}

{\small \hyperref{Trigonometry}{}{}{TR_contents.tex}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Angular Measure}{}{}{../../Help/DMIndex.tex#Angular measure%
}}

{\small \hyperref{Conic Sections}{}{}{../../Help/DMIndex.tex#Conic sections}}

{\small \hyperref{Plotting Curves and Surfaces}{}{}{../../Help/DM6.tex}}

{\small \hyperref{Polygons}{}{}{../../Help/DMIndex.tex#Polygons}}

{\small \hyperref{Surfaces of Revolution}{}{}{%
../../Help/DMIndex.tex#Surfaces of revolution}}

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

\noindent \hrulefill

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\noindent \hrulefill

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