Mathematrix: Vortrag/ Zusammengesetzte Figuren

Drücken Sie den dunkel Flächeninhalt durch die Länge a der Seite des Quadrats aus!

Geometrie der Ebene

Name	Figur (Form)	Umfang	Fläche	Andere Formeln
Allgemeines Dreieck		${\displaystyle u=a+b+c}$	${\displaystyle A={\frac {a\cdot h_{a}}{2}}={\frac {b\cdot h_{b}}{2}}={\frac {c\cdot h_{c}}{2}}}$
Rechtwinkeliges Dreieck		${\displaystyle u=a+b+c}$	${\displaystyle A={\frac {a\cdot b}{2}}}$	${\displaystyle c^{2}={a^{2}+b^{2}}}$ ${\displaystyle c={\sqrt {a^{2}+b^{2}}}}$
Gleichseitiges Dreieck		${\displaystyle u=3a}$	${\displaystyle A={\frac {\sqrt {3}}{4}}\cdot a^{2}}$	${\displaystyle h={\frac {\sqrt {3}}{2}}a}$
Rechteck		${\displaystyle u=2a+2b}$ ${\displaystyle u=2(a+b)}$	${\displaystyle A={a\cdot b}}$	${\displaystyle d^{2}={a^{2}+b^{2}}}$ ${\displaystyle d={\sqrt {a^{2}+b^{2}}}}$
Quadrat		${\displaystyle u=4a}$	${\displaystyle A={a^{2}}}$	${\displaystyle d={\sqrt {2}}a}$
Raute (Rhombus)		${\displaystyle u=4a}$	${\displaystyle A={\frac {e\cdot f}{2}}}$	${\displaystyle \textstyle {a^{2}=({\frac {e}{2}})^{2}+({\frac {f}{2}})^{2}}}$
Parallelogramm		${\displaystyle u=2a+2b}$ ${\displaystyle u=u=2(a+b)}$	${\displaystyle A={a\cdot h_{a}}={b\cdot h_{b}}}$
Trapez		${\displaystyle u=a+b+c+d}$	${\displaystyle A={\frac {a+c}{2}}\cdot h}$
Kreis		${\displaystyle u=2\pi \ r}$ ${\displaystyle u=\pi \ d}$	${\displaystyle A={\pi \ r^{2}}}$ ${\displaystyle A={\frac {\pi \ d^{2}}{4}}}$
Kreisring		${\displaystyle u=2\pi \ (R+r)}$	${\displaystyle A={\pi \ (R^{2}-r^{2})}}$

Geometrie des Raums

Name	Figur (Form)	Oberfläche	Volumen	Netz (falls möglich)
Würfel		${\displaystyle A_{O}=6a^{2}}$	${\displaystyle V=a^{3}}$
Quader		${\displaystyle A_{O}=2ab+2ac+2bc}$	${\displaystyle V=abc}$
Quadratische Pyramide		${\displaystyle {\begin{aligned}A_{O}&=A_{G}&&+A_{M}\\&=a^{2}&&+2a\ h_{1}\end{aligned}}}$   ${\displaystyle \left[A_{O}=a(a+h_{1})\right]}$   wobei ${\displaystyle h_{1}={\sqrt {\left({\frac {a}{2}}\right)^{2}+h^{2}}}}$ ${\displaystyle h_{1}\ }$ mit ${\displaystyle a_{1}\ }$ im Bild	${\displaystyle V={\frac {a^{2}\ h}{3}}}$
Tetraeder		${\displaystyle A_{O}={\sqrt {3}}a^{2}}$	${\displaystyle V={\frac {{\sqrt {2}}a^{3}}{12}}}$
Zylinder		${\displaystyle {\begin{aligned}A_{O}&=2\ \ A_{G}&&+A_{M}\\&=2\ \ \pi r^{2}&&+2\pi r\ h\end{aligned}}}$   ${\displaystyle \left[A_{O}=2\pi r(r+h)\right]}$	${\displaystyle V=\pi r^{2}\ h}$
Kegel		${\displaystyle {\begin{aligned}A_{O}&=A_{G}&&+A_{M}\\&=\pi r^{2}&&+\pi r\ s\end{aligned}}}$   ${\displaystyle \left[A_{O}=\pi r(r+s)\right]}$   wobei ${\displaystyle s={\sqrt {r^{2}+h^{2}}}}$	${\displaystyle V={\frac {\pi r^{2}\ h}{3}}}$
Prisma[1]		${\displaystyle {\begin{aligned}A_{O}&=2\ \ A_{G}&&+A_{M}\\&=\ \ {3{\sqrt {3}}}{}a^{2}&&+6a\ h\end{aligned}}}$	${\displaystyle V=\overbrace {{\frac {3{\sqrt {3}}}{2}}a^{2}} ^{A_{G}}\ \ \cdot h}$
Kugel		${\displaystyle A_{O}=4\pi r^{2}}$	${\displaystyle \textstyle V={\frac {4}{3}}\pi r^{3}}$
Torus		${\displaystyle A_{O}=4\pi ^{2}rR\ }$	${\displaystyle V=2\pi ^{2}r^{2}R}$

1. ↑ mit regelmäßigem Sechseck als Basis

${\displaystyle A_{Q}=a^{2},\ \ A_{GD}={\tfrac {{\sqrt {3}}\ a^{2}}{4}}\ \ A_{K}=\pi \ r^{2}={\tfrac {\pi \ d^{2}}{4}}}$

Seite des Dreiecks=Seite des Quadrats=a

Durchmesser des Kreises= ${\displaystyle {\tfrac {1}{3}}\ }$ der Seite des Quadrats
${\displaystyle d={\tfrac {1}{3}}\ a}$

Radius des Kreises= ${\displaystyle {\tfrac {1}{6}}\ }$ der Seite des Quadrats
${\displaystyle r={\tfrac {1}{6}}\ a}$

${\displaystyle A=a^{2}-{\tfrac {{\sqrt {3}}\ a^{2}}{4}}-2\cdot \pi \ \left({\tfrac {a}{6}}\right)^{2}}$
${\displaystyle A=a^{2}-{\tfrac {{\sqrt {3}}\ a^{2}}{4}}-2\cdot \pi \ {\tfrac {\left({\tfrac {a}{3}}\right)^{2}}{4}}}$

Berechnen Sie die dunklen Flächen in der folgenden Figur!

${\displaystyle A_{Q}=a^{2},\ \ A_{D}={\tfrac {a\cdot h_{a}}{2}},\ \ A_{K}=\pi \ r^{2}}$

${\displaystyle A_{Q}=a^{2},\ \ A_{RD}={\tfrac {a\cdot b}{2}},\ \ A_{K}=\pi \ r^{2},\ \ A_{D}={\tfrac {a\cdot h_{a}}{2}}}$

Großes "unsichtbares" Quadrat: ${\displaystyle A_{GQ}=8^{2}}$
Großes Dreieck: ${\displaystyle A_{GD}=8^{2}-{\tfrac {8\cdot 2}{2}}-{\tfrac {6\cdot 6}{2}}-{\tfrac {8\cdot 2}{2}}=30\ {\text{Einheiten}}}$

Kleines Quadrat: ${\displaystyle A_{KQ}=3^{2}=9\ {\text{Einheiten}}}$
Kreis ${\displaystyle A_{K}=\pi \ 1{,}5^{2}}$
Kleines Dreieck: ${\displaystyle A_{KD}={\tfrac {3\cdot 1{,}5}{2}}=2{,}25\ {\text{Einheiten}}}$
Winzinges Quadrat: ${\displaystyle A_{WQ}=1^{2}=1\ {\text{Einheiten}}}$

${\displaystyle A=30-9+\pi \ 1{,}5^{2}-2{,}25+1=19{,}75+2{,}25\ \pi \approx 26{,}82\ {\text{Einheiten}}}$