SectorOfACircle 2002-11-23 05:23:19 2009-11-08 00:16:08 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{interior} \PMlinkescapeword{complete} A sector is a fraction of the interior of a circle, described by a central angle $\theta$. If $\theta = 2 \pi,$ the sector becomes a complete circle. \begin{center} \includegraphics{sector.eps} \end{center} If the central angle is $\theta,$ and the radius of the circle is $r,$ then the area of the sector is given by $$\frac{1}{2}r^2\theta$$ This is obvious from the fact that the area of a sector is $\frac{\theta}{2 \pi}$ times the area of the circle (which is $\pi r^2$). Note that, in the formula, $\theta$ is in radians. \textbf{Remark}. Since the length $a$ of the arc of the sector is $r\theta$, the area of the sector is $\frac{1}{2}ar$, which is equal to the area of a triangle with base $=a$ and the height $=r$.