Area of Regular Polygons

Area of Regular PolygonsBreak the figure down into trianglesNotice there are 6 triangles and 6 sides126543Lets look at 1 of the triangles closer126543Area = bhhbThe area of a triangle is:hbWhen the triangle is part of a polygon, the height is called the ApothemapothemapothembaseOf course, using the equation youll only find the area for one of the triangles. There are 6 triangles, so we have to multiply our area by 6!apothembaseRemember, 6 times the base is the perimeter of the object!Another way to look at it is that we have 6 bases and one apothemapothembaseTherefore we can use the formula:Area =12__apPerimeter85Ex: Find the area of the hexagonArea =12__apFirst we must find the perimeter:p = 5(6) = 3085Find the area of the hexagonArea =12__apPlug everything inp = 5(6) = 30Area =12__8(30)85Find the area of the hexagonSolveArea =12__8(30)= 120Practice 1Find the area of the following regular polygons:Octagon with base of 4 and apothem 7

Hexagon with base of 8 and apothem 3

The figure below

1012But what if were not given the apothem??Area =12__ap10Ex: Find the area of the polygonWe need the perimeter and apothemArea =12__apLets look at 1 of the triangles closer again12654310Inside we have two right trianglesa10We can find the angle at the top and use it to find the apothemaTheres 3600 around the center of the figureSince theres 6 sides we can find the angle at the top of each triangle by:3606= 60The top angle is 60 and the apothem cuts it in halfa600300Half the triangle means half the basea3005This is now a 30-60-90 right triangle!The apothem is:

10Ex: Find the area of the polygonWe need the perimeter and apothemArea =12__ap

10Ex: Find the area of the polygonArea =12__ap

But what if its not a special right triangle??Area =12__ap3605= 726Break it down to the small trianglea3603The angle gets cut in halfThe base is cut in halfUse the trig function to find apothema3603opphypadj