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Retvinklet trekant – Trigonometri (3:3). Beregning af arealet i en retvinklet trekant.

Artiklen er nummer tre i serien, og dermed den sidste. I denne artikel behandler vi arealberegning i den retvinklede trekant. Artiklen har til formål at gøre den studerende i stand til, at identificere og løse eksamensopgaver, hvor man skal beregne arealet i en retvinklet trekant.

Problemet for de studerende i relation til arealberegning i den retvinklede trekant er oftest, at den studerende har svært ved at gennemskue trekantens højde og grundlinje. Vi giver dig svarene her.

Beregning af arealet i en retvinklet trekant

Arealet af en retvinklet kan beregnes ved hjælp af formlen:

Areal trekant 1

Hvor h er trekantens højde og g er trekantens grundlinje. Herunder er vist en retvinklet trekant ABC;

Areal trekant 2

Trekantens højde er markeret ved den rødstiplede linje, hvilket betyder at højden h = 5. Trekantens grundlinje er vist som den blåstiplede linje, hvorfor grundlinje g = 8. Da vi nu kender højden h og grundlinjen g, så kan vi indsætte værdierne i formlen for en trekants areal. Det gøres herunder:

Areal trekant 3

 

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Vi har dermed beregnet den retvinklet trekants areal i ovenstående eksempel til 20. Hvis vi igen anvender ovenstående retvinklet trekant (med de samme mål), så den retvinklede trekants hypotenuse bliver lig grundlinjen, så vil vi nu frem til samme areal. Se trekanten herunder:

Areal trekant 4

I figuren ovenfor er trekantens højde lig med den vinkelrette sortstiplede linje fra punktet A, og vinkelret ned på linjestykket BC (hypotenusen). Højden er markeret som den rødstiplede linje og er 4,24.

Trekantens grundlinje er markeret som den blåstiplede linje og er lig med 9,43 (længden af linjestykket BC). For at beregne den retvinklede trekants areal, så indsætter vi i formlen for arealberegning. Det gøres herunder;

Areal trekant 5

Vi laver igen om på vores eksempel så højden = 8 og grundlinjen = 5. Se figuren herunder:

Areal trekant 6

Højden er som i de andre eksempler udtrykt ved den rødstiplede linje, og grundlinjen er markeret med den blåstiplede linje. Vi indsætter nu igen i arealformlen herunder;

Areal trekant 7

En anden måde til arealberegning (ved brug af Sinus)

Der er også en anden måde hvorpå man kan beregne en trekants areal. Man kan nemlig også benytte arealformlen for vilkårlige trekanter på retvinklede trekanter. Denne arealformel er vist herunder;

Areal trekant 8

Hvis man anvender denne formel vil man nå frem til samme resultat som tidligere. Lad os tage et eksempel. Vi anvender den retvinklede trekant fra tidligere, men vi laver om i vores eksempel så vi nu antager, at vi ikke kender siden AB (højden), men vi kender vinklen C = 32,01 grader, sidelængden BC = 9,43 og sidelængden AC = 8. Se figuren herunder;

Areal trekant 9

Da vi ikke kender højden kan vi ikke bruge den gængse formel til beregning af den retvinklede trekants areal. Vi kan dog anvende nedenstående formel i stedet:

Areal trekant 10

Men først defineres siderne trekanten;

Areal trekant 11

Vi indsætter nu de kendte værdier i formlen herunder for, at beregne trekantens areal;

Areal trekant 12

Hvis vi igen laver om på vores eksempel, så vi nu antager at vi grundlinjen, men vi kender vinkel B = 57,99, sidelængden BC = 9,43 og sidelængden AB = 5. Se nedenstående figur:

Areal trekant 13

Arealet af trekanten kan beregnes ved formlen:

Areal trekant 14

Men først defineres trekantens sidelængder:

Areal trekant 15

Vi indsætter nu i formlen:

Areal trekant 16

 

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Hvis vi laver om på vores eksempel endnu en gang, så vi ikke kender sidelængden BC, men vi kender vinkel A = 90 grader, sidelængden AB = 5 og sidelængden AC = 8. Se figuren herunder:

Areal trekant 17

Trekantens areal kan igen beregnes ved hjælp af Sinus arealformlen:

Areal trekant 18

Vi definerer dog trekantens sider først:

Areal trekant 19

Vi indsætter i arealformlen:

Areal trekant 20

Vi når altså frem til det samme resultat som ved at bruge den ”gængse” formel for beregning af arealet i en trekant. Forskellen er bare at man beregner arealet ved hjælp af to sidelængder og en vinkel, i stedet for trekantens højde og grundlinje.

Nu burde du være i stand til at beregne en trekants areal. Du må meget gerne kommentere artiklen.

Vi håber du finder den nyttig – rigtig god fornøjelse 🙂

Her kan du finde endnu flere relevante matematik artikler:

Lær matematik hurtigt – se her hvordan, Bliv god til potens funktioner – her lærer du hvordan!, Gode råd til matematik eksamen

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