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Mathematrix: MA TER/ Formelsammlung Geometrie
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Mathematrix: MA TER
Geometrie der Ebene
Name
Figur (Form)
Umfang
Fläche
Andere Formeln
Allgemeines Dreieck
u
=
a
+
b
+
c
{\displaystyle u=a+b+c}
A
=
a
⋅
h
a
2
=
b
⋅
h
b
2
=
c
⋅
h
c
2
{\displaystyle 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
{\displaystyle u=a+b+c}
A
=
a
⋅
b
2
{\displaystyle A={\frac {a\cdot b}{2}}}
c
2
=
a
2
+
b
2
{\displaystyle c^{2}={a^{2}+b^{2}}}
c
=
a
2
+
b
2
{\displaystyle c={\sqrt {a^{2}+b^{2}}}}
Gleichseitiges Dreieck
u
=
3
a
{\displaystyle u=3a}
A
=
3
4
⋅
a
2
{\displaystyle A={\frac {\sqrt {3}}{4}}\cdot a^{2}}
h
=
3
2
a
{\displaystyle h={\frac {\sqrt {3}}{2}}a}
Rechteck
u
=
2
a
+
2
b
{\displaystyle u=2a+2b}
u
=
2
(
a
+
b
)
{\displaystyle u=2(a+b)}
A
=
a
⋅
b
{\displaystyle A={a\cdot b}}
d
2
=
a
2
+
b
2
{\displaystyle d^{2}={a^{2}+b^{2}}}
d
=
a
2
+
b
2
{\displaystyle d={\sqrt {a^{2}+b^{2}}}}
Quadrat
u
=
4
a
{\displaystyle u=4a}
A
=
a
2
{\displaystyle A={a^{2}}}
d
=
2
a
{\displaystyle d={\sqrt {2}}a}
Raute (Rhombus)
u
=
4
a
{\displaystyle u=4a}
A
=
e
⋅
f
2
{\displaystyle A={\frac {e\cdot f}{2}}}
a
2
=
(
e
2
)
2
+
(
f
2
)
2
{\displaystyle \textstyle {a^{2}=({\frac {e}{2}})^{2}+({\frac {f}{2}})^{2}}}
Parallelogramm
u
=
2
a
+
2
b
{\displaystyle u=2a+2b}
u
=
u
=
2
(
a
+
b
)
{\displaystyle u=u=2(a+b)}
A
=
a
⋅
h
a
=
b
⋅
h
b
{\displaystyle A={a\cdot h_{a}}={b\cdot h_{b}}}
Trapez
u
=
a
+
b
+
c
+
d
{\displaystyle u=a+b+c+d}
A
=
a
+
c
2
⋅
h
{\displaystyle A={\frac {a+c}{2}}\cdot h}
Kreis
u
=
2
π
r
{\displaystyle u=2\pi \ r}
u
=
π
d
{\displaystyle u=\pi \ d}
A
=
π
r
2
{\displaystyle A={\pi \ r^{2}}}
A
=
π
d
2
4
{\displaystyle A={\frac {\pi \ d^{2}}{4}}}
Kreisring
u
=
2
π
(
R
+
r
)
{\displaystyle u=2\pi \ (R+r)}
A
=
π
(
R
2
−
r
2
)
{\displaystyle A={\pi \ (R^{2}-r^{2})}}
Geometrie des Raums
Name
Figur (Form)
Oberfläche
Volumen
Netz (falls möglich)
Würfel
A
O
=
6
a
2
{\displaystyle A_{O}=6a^{2}}
V
=
a
3
{\displaystyle V=a^{3}}
Quader
A
O
=
2
a
b
+
2
a
c
+
2
b
c
{\displaystyle A_{O}=2ab+2ac+2bc}
V
=
a
b
c
{\displaystyle V=abc}
Quadratische
Pyramide
A
O
=
A
G
+
A
M
=
a
2
+
2
a
h
1
{\displaystyle {\begin{aligned}A_{O}&=A_{G}&&+A_{M}\\&=a^{2}&&+2a\ h_{1}\end{aligned}}}
[
A
O
=
a
(
a
+
h
1
)
]
{\displaystyle \left[A_{O}=a(a+h_{1})\right]}
wobei
h
1
=
(
a
2
)
2
+
h
2
{\displaystyle h_{1}={\sqrt {\left({\frac {a}{2}}\right)^{2}+h^{2}}}}
h
1
{\displaystyle h_{1}\ }
mit
a
1
{\displaystyle a_{1}\ }
im Bild
V
=
a
2
h
3
{\displaystyle V={\frac {a^{2}\ h}{3}}}
Tetraeder
A
O
=
3
a
2
{\displaystyle A_{O}={\sqrt {3}}a^{2}}
V
=
2
a
3
12
{\displaystyle V={\frac {{\sqrt {2}}a^{3}}{12}}}
Zylinder
A
O
=
2
A
G
+
A
M
=
2
π
r
2
+
2
π
r
h
{\displaystyle {\begin{aligned}A_{O}&=2\ \ A_{G}&&+A_{M}\\&=2\ \ \pi r^{2}&&+2\pi r\ h\end{aligned}}}
[
A
O
=
2
π
r
(
r
+
h
)
]
{\displaystyle \left[A_{O}=2\pi r(r+h)\right]}
V
=
π
r
2
h
{\displaystyle V=\pi r^{2}\ h}
Kegel
A
O
=
A
G
+
A
M
=
π
r
2
+
π
r
s
{\displaystyle {\begin{aligned}A_{O}&=A_{G}&&+A_{M}\\&=\pi r^{2}&&+\pi r\ s\end{aligned}}}
[
A
O
=
π
r
(
r
+
s
)
]
{\displaystyle \left[A_{O}=\pi r(r+s)\right]}
wobei
s
=
r
2
+
h
2
{\displaystyle s={\sqrt {r^{2}+h^{2}}}}
V
=
π
r
2
h
3
{\displaystyle V={\frac {\pi r^{2}\ h}{3}}}
Prisma
[
1
]
A
O
=
2
A
G
+
A
M
=
3
3
a
2
+
6
a
h
{\displaystyle {\begin{aligned}A_{O}&=2\ \ A_{G}&&+A_{M}\\&=\ \ {3{\sqrt {3}}}{}a^{2}&&+6a\ h\end{aligned}}}
V
=
3
3
2
a
2
⏞
A
G
⋅
h
{\displaystyle V=\overbrace {{\frac {3{\sqrt {3}}}{2}}a^{2}} ^{A_{G}}\ \ \cdot h}
Kugel
A
O
=
4
π
r
2
{\displaystyle A_{O}=4\pi r^{2}}
V
=
4
3
π
r
3
{\displaystyle \textstyle V={\frac {4}{3}}\pi r^{3}}
Torus
A
O
=
4
π
2
r
R
{\displaystyle A_{O}=4\pi ^{2}rR\ }
V
=
2
π
2
r
2
R
{\displaystyle V=2\pi ^{2}r^{2}R}
↑
mit regelmäßigem
Sechseck
als Basis