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Beregn areal og vinkel i en vilkårlig trekant – Læs og forstå det på under 2 minutter!

Vi kender alle fra folkeskolen, hvordan man finder arealet af en retvinklet trekant. Hvor man blot skal gange grundlinjen med højden og til sidst gange med en halv – så har du fundet arealet.

Men uheldigt for os, så er det de færreste trekanter vi bliver udsat for, som der er retvinklede. Langt de fleste er faktisk vilkårlige trekanter. Derfor vil jeg med denne artikel gennemgå, hvordan du beregner arealet og en vinkel i en vilkårlig trekant.

Vi har en formel, der kan give os arealet af en vilkårlig trekant, blot vi kender de rette vinkler og sider. Har man givet en vilkårlig trekant ABC

Så kan arealet at trekanten bestemmes ved formlen

Man skal altså kende en vinkel og vinklens to hosliggende sider for at kunne bestemme arealet. Dette betyder at arealet også er givet ved

Nu har vi fået bestemt hvad arealsætningen for vilkårlige trekanter er, og det næste skridt er at gennemgå 2 eksempler, så du lærer at beregne arealet af vilkårlige trekanter og finde vinkler i vilkårlige trekanter.

 

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Beregning af arealet i en vilkårlig trekant

Vi får givet den vilkårlige trekant ABC, hvor vinkel B = 38 grader, siden a = 8  og
siden c = 11 . Vi vil gerne bestemme arealet af trekanten.

Vi starter med at tegne trekanten herunder

Vi skal altså bestemme arealet af trekanten.

Vi kender en vinkel og dens hosliggende sider. Vi kan derfor bestemme arealet ved blot at benytte arealformlen. Vi opskriver derfor

Vi indsætter nu de værdier vi kender

Vi udregner nu ovenstående og dermed får vi arealet til

Vi har nu vist at vi kan benytte arealsætningen til at bestemme arealet af en vilkårlig trekant.

 

 

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Beregning af vinkel i en vilkårlig trekant

I dette eksempel vil jeg nu gennemgå, hvordan du beregner vinklen, hvis du har opgivet arealet og 2 hosliggende sider. Det er lidt som den omvendte øvelse, og denne type opgave kan nemt komme til matematik eksamenen.

Vi får givet den vilkårlige trekant ABC, hvor arealet er 30, siden b = 10 og siden c = 12.
Vi vil gerne bestemme vinklen.

Vi starter med at tegne trekanten

Vi skal altså bestemme vinkel A.

Vi kender arealet og vinklens hosliggende sider. Vi kan derfor bestemme vinklen ved blot at benytte arealformlen. Vi opskriver derfor

Vi indsætter nu de værdier vi kender

Vi skal som sagt bestemme vinklen , vi starter derfor med at isolere

Vi udregner nu højre siden af lighedstegnet

Vi kan nu bestemme vinklen  ved at tage sinus i minus første på begge sider af lighedstegnet

Vi udregner nu udtrykket og dermed fås vinkel  til

VI har nu vist at vi også kan bestemme en vinkel, når vi kender arealet vinklens to hosliggende sider.

Det var de 2 gode eksempler jeg havde til dig, så du nu forhåbentligt har fået helt styr på areal og vinkel beregning i en vilkårlig trekant. Jeg har herunder samlet flere gode links til dig, hvis du skulle have lyst til at lære mere om den retvinklede trekant 🙂

Trigonometri – retvinklet trekant (1:3). Beregning af sidelængder ved Pythagoras.

Retvinklet trekant – Trigonometri (2:3). Beregning af sidelængder og vinkler ved brug af Sinus, Cosinus og Tangens.

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

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