Mathematrix: MA TER/ Formelsammlung Geometrie

Geometrie der Ebene

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

Geometrie des Raums

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