Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of trigonometry wikipedia , lookup
Positional notation wikipedia , lookup
Functional decomposition wikipedia , lookup
Large numbers wikipedia , lookup
Elementary algebra wikipedia , lookup
Location arithmetic wikipedia , lookup
Numerical continuation wikipedia , lookup
Approximations of π wikipedia , lookup
Mathematics and architecture wikipedia , lookup
Pythagorean theorem wikipedia , lookup
AreaofCommonShapes AreaofRectangle Figure1:A4-by-9rectanglewith36unitsquaresinside Ashape'sareameansthenumberof1-by-1squareunittheshapecovers.InFigure1,thisrectangle's areais 9 ⋅ 4 = 36 squareunits,astherectanglecovers361-by-1unitsquares.It'sfairlyeasyto understandthatarectangle'sareaformulais: rectanglearea=base·height(orlength·width) Areaoftriangle Figure2:Atriangle'sareaishalfasbigasarectanglewiththesamebaseandheight Bythisgraph,it'sfairlyeasytoseewhyatriangle'sareaishalfasbigastheareaofarectanglewiththe samebaseandsameheight.Thusatriangle'sareaformulais: 1 triangle area = (base)(height) 2 [Example1]Atriangle'sbaseis4meters,anditsheightis2.5meters.Finditsarea. [Solution]Bythetriangleareaformula,theareais: A= 1 1 (base)(height ) = ⋅ 4 ⋅ 2.5 = 5squaremeters 2 2 Noticethattheunitofareaisdifferentfromtheunitofperimeter.Ifarectangle'sperimeterissome meters,thenitsareamustbesomesquaremeters. Forsimplicity,wecanalsowrite"5squaremeters"as"5m2".Theletter"m"represents"meters". Similarly,"cm"representscentimeters,"in"representsinches,etc. Noticethat"5m2"isdifferentfrom"52m": • • "5m2"meansanareaof5squaremeters.Thesquarehasnothingtodowith5. "52m"meansalengthof25meters.Thesquarehasnothingtodowith"m". Arighttriangle'sheightisactuallyoneofitslegs,andanobtusetriangle'sheightliesoutsidethetriangle. Seethefollowingfigures: Figure3:righttriangle'sheightandobtusetriangle'sheight Inthenextexample,youneedtosolveanequationbasedonatriangle'sareaformula. [Example2]Atrianglecovers20squaremillimeters.Itsbaseis5millimeters.Finditsheight. [Solution]Letthetriangle'sheightbehmillimeters.Plugthegivennumbersintothetrianglearea formula,wehave: 1 (base )(height ) 2 1 20 = ⋅ 5h 2 1 2 ⋅ 20 = 2 ⋅ ⋅ 5h 2 40 = 5h A= 40 5h = 5 5 8=h Solution:Thetriangle'sheightis8millimeters.Notethatinthethirdrow,wemultipledbothsidesofthe equationby2togetridofthefraction 1 . 2 AreaofCirlces Next,let'slearnthefamouscircleareaformula: A = πr 2 .Theonlythingweneedtobecarefulaboutis thatsometimesthediameterisgiven,andweneedtofindtheradiusfirst. [Example3]Acircle'sdiameteris8yards.Findthiscircle'sareaintermsofπ,andthenroundtheareato thehundredthplace. [Solution]Wefirstfindthecircle'sradius,whichishalfofitsdiameter―4yardsinthisproblems.Next, weuseacircle'sareaformula: A = πr 2 = π ⋅ 4 2 = 16π yd2 Next,weuseascientificcalculatortochangetheresulttodecimal,andthenroundtothehundredth place: A = 16π = 16 ⋅ 3.1415026... ≈ 50.27 yd2 Solution:Thecircle'sareais16πyd2(accuratevalue),orapproximately50.27yd2. Inthenextexample,theareaisgiven,andyouareaskedtofindthecircle'sradius.Weneedtoreview theconceptof"squareroot"first. 02 = 0 0 =0 12 = 1 1 =1 22 = 4 4 =2 2 3 =9 9 =3 2 4 = 16 16 = 4 ... ... 10 2 = 100 100 = 10 Squarerootdoestheoppositeofsquare.Ifweknow r 2 = 100 ,weusesquareroottofindr'svalue: r 2 = 100 r = 100 r = 10 [Example4]Acircle'sareais100squaremeters.Findthisradiusdiameter.Roundyouranswerto hundredthplace. [Solution]Tofindacircle'sdiameter,weneedtofinditsradius.Plug A = 100 intoacircle'sarea formula,wehave: A = πr 2 100 = πr 2 100 π 100 π 100 π = πr 2 π = r2 =r 5.64 ≈ r Sinceacircle'sdiameteristwiceitsradius,wehave: d = 2r = 2 ⋅ 5.64 = 11.28 meters Solution:Thecircle'sdiameterisapproximately11.28meters.