Geometrie - Grundlagen
Geometrie - Dreieck
- Allgemeines Dreieck
- $A = \frac{g \cdot h}{2}$
- $g = \frac{A\cdot 2}{ h}$
- $h = \frac{A\cdot 2}{ g}$
- $A =\frac{1}{2}\cdot a\cdot b\cdot sin(\gamma)$
- $U = a+b+c$
- Gleichseitiges Dreieck
- $A = \frac{a^{2} }{4}\cdot \sqrt{3}$
- $a = \sqrt{\frac{A\cdot 4}{\sqrt{3}}}$
- $h = \frac{a}{2}\cdot \sqrt{3}$
- $a = \frac{h\cdot 2}{\sqrt{3}}$
- Rechtwinkliges Dreieck
- $A = \frac{a\cdot b}{ 2}$
- $a = \frac{A \cdot 2}{ b}$
- $b = \frac{A \cdot 2}{ a}$
- $c =\sqrt{a^{2} + b^{2} }$
- $a =\sqrt{c^{2} - b^{2} }$
- $b =\sqrt{c^{2} - a^{2} }$
- $h = \sqrt{p\cdot q}$
- $q = \frac{h^{2} }{p}$
- $p = \frac{h^{2} }{q}$
- $a = \sqrt{c\cdot p}$
- $c = \frac{a^{2} }{p}$
- $p = \frac{a^{2} }{c}$
- Gleichschenkliges rechtwinkliges Dreieck
- $A = \frac{a\cdot b}{ 2}$
- $a = \frac{A \cdot 2}{ b}$
- $b = \frac{A \cdot 2}{ a}$
- $c =\sqrt{a^{2} + b^{2} }$
- $a =\sqrt{c^{2} - b^{2} }$
- $b =\sqrt{c^{2} - a^{2} }$
- $h = \sqrt{p\cdot q}$
- $q = \frac{h^{2} }{p}$
- $p = \frac{h^{2} }{q}$
- $a = \sqrt{c\cdot p}$
- $c = \frac{a^{2} }{p}$
- $p = \frac{a^{2} }{c}$
Geometrie - Viereck
- Quadrat
- $A = a^{2}$
- $a = \sqrt{A}$
- $U = 4\cdot a$
- $a = \frac{U}{4}$
- $d = a\cdot \sqrt{2}$
- $a = \frac{d}{\sqrt{2}}$
- Rechteck
- $A = a\cdot b$
- $a = \frac{A}{b}$
- $b = \frac{A}{a}$
- $U = 2\cdot a + 2\cdot b$
- $a = \frac{U - 2\cdot b}{ 2}$
- $b = \frac{U - 2\cdot a}{ 2}$
- $d = \sqrt{a^{2} +b^{2} }$
- $b = \sqrt{d^{2} -a^{2} }$
- $a = \sqrt{d^{2} -b^{2} }$
- Allgemeines Trapez
- $A = \frac{a+c}{ 2}\cdot h$
- $a = \frac{2\cdot A}{ h} - c$
- $c = \frac{2\cdot A}{ h} - a$
- $h = \frac{2\cdot A}{a+c}$
- Gleichschenkliges Trapez
- $A = \frac{a+c}{ 2}\cdot h$
- $a = \frac{2\cdot A}{ h} - c$
- $c = \frac{2\cdot A}{ h} - a$
- $h = \frac{2\cdot A}{a+c}$
- Rechtwinkliges Trapez
- $A = \frac{a+c}{ 2}\cdot h$
- $a = \frac{2\cdot A}{ h} - c$
- $c = \frac{2\cdot A}{ h} - a$
- $h = \frac{2\cdot A}{a+c}$
Geometrie - Polygone (n-Ecken)
- Sechseck
- $A = \frac{3 \cdot a^{2} }{2}\cdot \sqrt{3}$
- $a = \sqrt{\frac{A\cdot 2}{3 \cdot \sqrt{3}}}$
- $\rho= \frac{a}{2}\cdot \sqrt{3}$
- $a = \frac{\rho \cdot 2}{\sqrt{3}}$
Geometrie - Kreis
- Kreis
- $d= 2\cdot r$
- $r= \frac{d}{2}$
- $A = r^{2} \cdot \pi$
- $r = \sqrt{\frac{A}{\pi }}$
- $U = 2\cdot r\cdot \pi$
- $r = \frac{ U}{2\cdot \pi }$
- Kreissektor (Grad)
- $A = \frac{r^{2} \cdot \pi \cdot \alpha }{ 360}$
- $r = \sqrt{\frac{A\cdot 360}{\alpha \cdot \pi }}$
- $\alpha = \frac{A\cdot 360}{r^{2} \cdot \pi }$
- $b = \frac{2\cdot r\cdot \pi \cdot \alpha }{ 360}$
- $r = \frac{b\cdot 360}{\alpha \cdot \pi \cdot 2}$
- $\alpha = \frac{b\cdot 360}{r\cdot \pi \cdot 2}$
- Kreissektor (Bogenmaß)
- $A = \frac{r^{2} \cdot x }{ 2}$
- $r = \sqrt{\frac{A\cdot 2}{x }}$
- $x = \frac{A\cdot 2}{r^{2} }$
- $b = r\cdot x$
- $r = \frac{b}{x }$
- $x = \frac{b}{r}$
- Kreisring
- $A = (r_{a} ^{2} - r_{i} ^{2} )\cdot\pi$
- $r_{a} = \sqrt{\frac{A}{\pi } + r_{i} ^{2} }$
- $r_{i} = \sqrt{r_{a} ^{2} - \frac{A}{\pi } }$
Geometrie - Stereometrie
- Prisma
- $V = G\cdot h$
- $G = \frac{V}{h}$
- $h = \frac{V}{G}$
- $O = 2\cdot G +M $
- $G = \frac{O-M}{2}$
- $M = O- 2\cdot G $
- Würfel
- $V = a^{3}$
- $a = ^{3} \sqrt{V}$
- $O = 6\cdot a^{2}$
- $a = \sqrt{\frac{O}{6}}$
- $d = a\cdot \sqrt{3}$
- $a = \frac{d}{\sqrt{3}}$
- Quader
- $V = a\cdot b\cdot c$
- $a = \frac{ V}{b\cdot c}$
- $b = \frac{ V}{a\cdot c}$
- $c = \frac{ V}{b\cdot a}$
- $O = 2\cdot (a\cdot b + a\cdot c + b\cdot c)$
- $a = \frac{O-2\cdot b\cdot c}{2\cdot (b+c)}$
- $b = \frac{O-2\cdot a\cdot c}{2\cdot (a+c)}$
- $c = \frac{O-2\cdot b\cdot a}{2\cdot (b+a)}$
- $d = \sqrt{a^{2} +b^{2} +c^{2} }$
- $a = \sqrt{d^{2} -b^{2} -c^{2} }$
- $b = \sqrt{d^{2} -a^{2} -c^{2} }$
- $c = \sqrt{d^{2} -b^{2} -a^{2} }$
- Pyramide
- $V =\frac{1}{3} G\cdot h$
- $G = \frac{3 \cdot V}{h}$
- $h = \frac{3 \cdot V}{G}$
- $O = G +M $
- $G = O-M$
- $M = O- G $
- $\text{Rechteckige Pyramide}$
- $\text{Quadratische Pyramide}$
- Kreiszylinder
- $V = r^{2} \cdot \pi \cdot h$
- $r = \sqrt{\frac{ V}{\pi \cdot h}}$
- $h = \frac{ V}{r^{2} \cdot \pi }$
- $O = 2\cdot r\cdot \pi \cdot (r+h)$
- $r = 0,5\cdot (-h+\sqrt{h^{2} +\frac{O}{\pi }})$
- $h = \frac{0-2\cdot \pi \cdot r^{2} }{ 2\cdot r\cdot \pi }$
- Hohlzylinder
- $V = (r_{1} ^{2} - r_{2} ^{2} )\cdot \pi \cdot h$
- $r_{1} = \sqrt{\frac{ V}{\pi \cdot h}+r_{2} ^{2} }$
- $r_{2} = \sqrt{r_{1} ^{2} - \frac{ V}{\pi \cdot h}}$
- $h = \frac{ V}{(r_{1} ^{2} - r_{2} ^{2} )\cdot \pi }$
- Kreiskegel
- $V = \frac{1}{3}\cdot r^{2} \cdot \pi \cdot h$
- $r = \sqrt{\frac{3\cdot V}{\pi \cdot h}}$
- $h = \frac{3\cdot V}{r^{2} \cdot \pi }$
- $O = r\cdot \pi \cdot (r+s)$
- $s = \frac{ O}{r\cdot \pi } - r$
- $r = \frac{-\pi \cdot s + \sqrt{(\pi \cdot s)^{2} +4\cdot \pi \cdot O}}{ 2\cdot \pi }$
- $M = r\cdot \pi \cdot s$
- $s = \frac{ M}{r\cdot \pi }$
- $r = \frac{ M}{s\cdot \pi }$
- $s =\sqrt{h^{2} + r^{2} }$
- $r =\sqrt{s^{2} - h^{2} }$
- $h =\sqrt{s^{2} - r^{2} }$
- Kugel
- $V = \frac{4}{3}\cdot r^{3} \cdot \pi$
- $r = ^{3} \sqrt{\frac{V\cdot 3}{4\cdot \pi }}$
- $O = 4\cdot r^{2} \cdot \pi$
- $r = \sqrt{\frac{ O}{\pi \cdot 4}}$
Geometrie - Trigonometrie
- Definition
- $\sin \alpha - \cos \alpha - \tan \alpha $
- $\sin \alpha = y $
- $\cos \alpha = x $
- $\tan \alpha = m $
- Quadrantenregel
- $\sin \alpha - \cos \alpha - \tan \alpha $
- $\sin \alpha = y $
- $\cos \alpha = x $
- $\tan \alpha = m $
- Umrechnungen
- $sin \alpha = \sqrt{1 - cos^{2} \alpha }$
- $cos \alpha = \sqrt{1 - sin^{2} \alpha }$
- $tan \alpha = \frac{sin \alpha }{cos \alpha }$
- $sin \alpha = tan\alpha \cdot cos \alpha$
- $cos \alpha = \frac{sin \alpha }{tan \alpha }$
- Rechtwinkliges Dreieck
- $sin \alpha = \frac{a}{c}$
- $a = sin \alpha \cdot c$
- $c = \frac{ a}{sin\alpha }$
- $cos \alpha = \frac{b}{c}$
- $b = cos \alpha \cdot c$
- $c = \frac{ b}{cos \alpha }$
- $tan \alpha = \frac{a}{b}$
- $a = tan \alpha \cdot b$
- $b = \frac{ a}{tan\alpha }$