CK12 - Geometry

Thinkaboutthe area of a polygon.Imaginethat you look at a squarewith an area of exactly squareunit.

Of course,the sidesof the squareare unit of lengthlong.Now thinkaboutanotherpolygonthat is similar

to the first one with a scalefactorof. Every -by- squarein the first polygonhas a matching -by-

squarein the secondpolygon,and the area of eachof these -by- squaresis. Extendingthis

reasoning,every squareunit of area in the first polygonhas a corresponding units of area in the second

polygon.So the total area of the secondpolygonis timesthe area of the first polygon.

Warning:In solvingproblemsit’s easyto forgetthat you do not alwaysuse just the scalefactor. Use the
scalefactorin problemsaboutlengths. But use thesquareof the scalefactorin problemsaboutarea!

Example 2

Wu and Tomi are paintingmuralson rectangularwalls.The lengthand widthof Tomi’s wall are timesthe
lengthand widthof Wu’s wall.

a.The total lengthof the borderof Tomi’s wall is feet. Whatis the total lengthof the borderof Wu’s
wall?

This is a questionaboutlengths,so you use the scalefactoritself.All the sidesof Tomi’s wall are times

the lengthof the correspondingside of Wu’s wall,so the perimeterof Tomi’s wall is also timesthe
perimeterof Wu’s wall.

The total lengthof the border(perimeter)of Wu’s wall is feet.

b.Wu can coverhis wall with quartsof paint.How manyquartsof paintwill Tomi needto coverher wall?

This questionis aboutarea,sincethe area determinesthe amountof paintneededto coverthe walls.The
ratio of the amountsof paintis the sameas the ratio of the areas(whichis the squareof the scalefactor).
Let be the amountof paintthat Tomi needs.

Tomi wouldneed quartsof paint.

Summaryof Lengthand AreaRelationshipsfor SimilarPolygons

If two similarpolygonsare relatedby a scalefactorof , then:

• Length:The lengthsof any correspondingpartshavethe sameratio,. Notethat this appliesto sides,

*Area:The ratio of the areasis. Notethat this appliesto areas,and any aspectof an objectthat

Note:You mightbe able to makea prettygoodguessaboutthe volumesof similarsolid figures.
You’ll see moreaboutthat in Chapter11.

ScaleDrawingsand ScaleModels

One importantapplicationof similarfiguresis the use of scaledrawingsand scalemodels.Theseare two-
dimensional(scaledrawings)or three-dimensional(scalemodels)representationsof real objects.The
drawingor modelissimilarto the actualobject.