- Relatearea and perimeterformulasfor regularpolygonsto the limit processin prior lessons.
Introduction
You’veprobablybeenaskingyourself,“Wheredid the areasand perimetersof regularpolygonsin earlier
lessonscomefrom?”Or maybenot! You mightbe confidentthat the informationpresentedthen was accurate.
In eithercase,in this lessonwe’ll fill in the missinglink. We’ll deriveformulasfor the perimeterand area of
any regularpolygon.
You alreadyknowhow to find areasand perimetersof somefigures—triangles,rectangles,etc. Not surpris-
ingly, the new formulasin this lessonwill buildon thosebasicfigures—inparticular, the triangle.Notetoo
that we will find an outstandingapplicationof trigonometricfunctionsin this lesson.
Partsand Termsfor RegularPolygons
Let’s start with somebackgroundon regularpolygons.
Hereis a generalregularpolygonwith sides;someof its sidesare shown.
In the diagram,here is whateachvariablerepresents.
is the lengthof eachside of the polygon.is the lengthof a “radius”of the polygon,whichis a segmentfrom a vertexof the polygonto the
center.is the lengthof one-halfof a side of the polygon
is the lengthof a segmentcalledtheapothem—a segmentfrom the centerto a side of the polygon,
perpendicularto the side.(Noticethat is the altitudeof eachof the trianglesformedby two radii
and a side.)The anglebetweentwo consecutiveradii measures because congruentcentralanglesare formed
by the radii fromthe centerto eachof the verticesof the polygon.An apothemdivideseachof these