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Zeile 1: |
| {{TOC-Man|[[#Allgemeine Eigenschaften|Allgemeine Eigenschaften]] - [[#Satz des Pythagoras|Satz des Pythagoras]] - [[#Kathetensatz|Kathetensatz]] - [[#Höhensatz|Höhensatz]] - [[#Winkelfunktionen|Winkelfunktionen]] - [[#Sinussatz|Sinussatz]] - [[#Kosinussatz|Kosinussatz]] - [[#Fläche/Umfang|Fläche/Umfang]]}}
| | ein dreieck hat drei ecken. drei seiten. drei kanten. drei winkel. drei strecken. drei punkte. das ist ein dreieck. :] |
| == Allgemeine Eigenschaften ==
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| [[Bild:Dreieck.png|thumb|200px|Ein allgemeines Dreieck]]
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| {|
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| !|<div align="left">Bennennung der Eckpunkte</div>
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| |A, B und C (normalerweise im Uhrzeigersinn)
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| |-
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| !|<div align="left">[[Winkel]]summe</div>
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| |180°
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| |-
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| !|<div align="left">Bennennung der Winkel</div>
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| |α (am Eckpunkt A), β (am Eckpunkt B) und γ (am Eckpunkt C)
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| |-
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| !|<div align="left">Benennung der Seiten</div>
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| |Die Seiten werden nach dem Eckpunkt benannt mit Kleinbuchstaben, die ihnen gegenüber liegen.
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| |-
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| |}
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| == Satz des Pythagoras ==
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| 1+1=2 :D
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| http://www.science-at-home.net/bilder/wikidata/mathematik/rechtwinkliges_dreieck_01.gif
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''c²''' = a² + b²
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''a²''' = c² - b²
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''b²''' = c² - a²
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| |}
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| |}
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''c''' = √(a² + b²)
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''a''' = √(c² - b²)
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''b''' = √(c² - a²)
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| |}
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| |}
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| a Kathete
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| b Kathete
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| c Hypotenuse
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| == Kathetensatz ==
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| [[Bild:Kathetensatz.png|thumb|160px|Der Kathetensatz des Euklid]]
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| Das Quadrat über a ist flächengleich zum Rechteck mit den Seiten p und c, und das Quadrat über b ist flächengleich zum Rechteck mit den Seiten q und c.
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| {{Formel|=a<sup>2</sup> = p·c}}
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| {{Formel|=b<sup>2</sup> = q·c}}
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| == Höhensatz ==
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| {|
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| |[[Bild:Höhensatz.png|float:right|160px|Der Höhensatz des Euklid]]
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| |Der Höhensatz besagt, dass in einem rechtwinkligen Dreieck das Quadrat über der Höhe flächengleich dem Rechteck aus den Hypotenusenabschnitten ist.
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| |}
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| {{Formel|=h<sup>2</sup> = p·q}}
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| == Winkelfunktionen ==
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| http://www.science-at-home.net/bilder/wikidata/mathematik/rechtwinkliges_dreieck_02.gif
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''sin α''' = a : c
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''cos α''' = b : c
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''tan α''' = a : b
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''cot α''' = b : a
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| |}
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| |}
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''sin β''' = b : c
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''cos β''' = a : c
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''tan β''' = b : a
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''cot β''' = a : b
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| |}
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| |}
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| a Gegenkathete von α, Ankathete von β
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| b Gegenkathete von β, Ankathete von α
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| c Hypotenuse
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| == Sinussatz ==
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |a : b = sin α : sin β
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''a''' = (sin α · b) / sin β
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''b''' = (sin β · a) / sin α
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''sin α''' = (sin β · a) / b
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''sin β''' = (sin α · b) / a
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| |}
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| |}
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |b : c = sin β : sin γ
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''b''' = (sin β · c) / sin γ
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''c''' = (sin γ · b) / sin β
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''sin β''' = (sin γ · b) / c
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''sin γ''' = (sin β · c) / b
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| |}
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| |}
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |c : a = sin γ : sin α
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''c''' = (sin γ · a) / sin α
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''a''' = (sin α · c) / sin γ
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''sin γ''' = (sin α · c) / a
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''sin α''' = (sin γ · a) / c
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| |}
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| |}
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| == Kosinussatz ==
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''a²''' = b² + c² - 2bc · cos α
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''b²''' = a² + c² - 2ac · cos β
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''c²''' = a² + b² - 2ab · cos γ
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| |}
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| == Fläche/Umfang ==
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| http://www.science-at-home.net/bilder/wikidata/mathematik/dreieck.gif
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''A''' = (l<sub>1</sub> · b) / 2
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''l<sub>1</sub>''' = (A · 2) / b
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''b''' = (A · 2) / l<sub>1</sub>
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| |}
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| |}
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| {|
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#e1e8f2" align="center" valign="middle"
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| |'''U''' = l<sub>1</sub> + l<sub>2</sub> + l<sub>3</sub>
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''l<sub>1</sub>''' = U - (l<sub>2</sub> + l<sub>3</sub>)
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''l<sub>2</sub>''' = U - (l<sub>1</sub> + l<sub>3</sub>)
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| |}
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| {| border="1" cellpadding="4" cellspacing="0" style="BORDER-COLLAPSE: collapse" borderColor="#336699" align="center"
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| |- bgcolor="#edecf2" align="center" valign="middle"
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| |'''l<sub>3</sub>''' = U - (l<sub>1</sub> + l<sub>2</sub>)
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| |}
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| |}
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| A Fläche
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| l<sub>1</sub>, l<sub>2</sub>, l<sub>3</sub> Längen der Seiten
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| b Breite
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| U Umfang
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| ----
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| '''Pfad: [[Hauptseite|Home]] / [[Mathematik]] / [[Formeln Geometrie|Geometrie]] / [[Formeln Dreieck|Dreieck]]'''
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| [[Kategorie:Mathematik]]
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| [[Kategorie:Geometrie]]
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| [[Kategorie:Formeln]]
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