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Figuras de dos dimensiones [ editar ]
Nombre
Área interior
Perímetro
Lados
Vértices
Comentarios
Dibujo
Triángulo
b
h
2
{\displaystyle {\frac {bh}{2}}}
a
+
b
+
c
{\displaystyle a+b+c\,}
3
{\displaystyle 3\,}
3
{\displaystyle 3\,}
b base , h altura , a c Triángulo
Triángulo equilátero
3
4
a
2
{\displaystyle {\frac {\sqrt {3}}{4}}a^{2}}
3
a
{\displaystyle 3a}
3
{\displaystyle 3\,}
3
{\displaystyle 3\,}
a
Cuadrado
a
2
{\displaystyle a^{2}\,}
4
a
{\displaystyle 4a}
4
{\displaystyle 4\,}
4
{\displaystyle 4\,}
a Cuadrados
Rombo
A
C
¯
⋅
B
D
¯
2
{\displaystyle {\frac {{\overline {AC}}\cdot {\overline {BD}}}{2}}}
4
a
{\displaystyle 4a\,}
4
{\displaystyle 4\,}
4
{\displaystyle 4\,}
a AC diagonal menor, y BD diagonal mayor
Rombo
Rectángulo
b
h
{\displaystyle bh}
2
(
b
+
h
)
{\displaystyle 2(b+h)}
4
{\displaystyle 4\,}
4
{\displaystyle 4\,}
b h altura
Paralelogramo
b
h
{\displaystyle bh}
2
(
a
+
b
)
{\displaystyle 2(a+b)}
4
{\displaystyle 4\,}
4
{\displaystyle 4\,}
b a h altura
Paralelogramo
Trapecio
(
a
+
c
)
h
2
{\displaystyle {\frac {(a+c)h}{2}}}
a
+
b
+
c
+
d
{\displaystyle \ a+b+c+d\,}
4
{\displaystyle 4\,}
4
{\displaystyle 4\,}
a c h altura , b d
Pentágono regular
25
+
10
5
4
a
2
{\displaystyle {\frac {\sqrt {25+10{\sqrt {5}}}}{4}}a^{2}}
5
a
{\displaystyle 5a}
5
{\displaystyle 5\,}
5
{\displaystyle 5\,}
a
Polígono regular
n
a
b
2
{\displaystyle {\frac {nab}{2}}}
n
⋅
a
{\displaystyle n\cdot a}
n
{\displaystyle n\,}
n
{\displaystyle n\,}
a b apotema del polígono, y n
Polígono regular
n
2
sen
(
2
π
n
)
r
2
{\displaystyle {\frac {n}{2}}\operatorname {sen} \left({\frac {2\pi }{n}}\right)r^{2}}
n
⋅
a
{\displaystyle n\cdot a}
n
{\displaystyle n\,}
n
{\displaystyle n\,}
a r n radianes .
Figura cerrada por una curva [ editar ]
Nombre
Área
Perímetro
Comentarios
Círculo
π
r
2
{\displaystyle \pi r^{2}}
2
π
r
{\displaystyle 2\pi r}
donde
r
{\displaystyle r\,}
Elipse
π
r
1
r
2
{\displaystyle \pi r_{1}r_{2}}
P
≈
π
[
3
(
r
1
+
r
2
)
−
(
3
r
1
+
r
2
)
(
r
1
+
3
r
2
)
]
{\displaystyle P\approx \pi \left[3(r_{1}+r_{2})-{\sqrt {(3r_{1}+r_{2})(r_{1}+3r_{2})}}\right]\!\,}
r
1
{\displaystyle r_{1}\,}
semieje , y
r
2
{\displaystyle r_{2}\,}
otro
Cardioide
6
π
r
2
{\displaystyle 6\pi r^{2}}
16
r
{\displaystyle 16r}
r
{\displaystyle r\,}
ecuaciones paramétricas del cardioide.
Figuras de tres dimensiones [ editar ]
Nombre
Volumen
Superficie
Caras
Aristas
Vértices
Comentarios
Dibujo
Cubo
a
3
{\displaystyle a^{3}\,}
6
a
2
{\displaystyle 6a^{2}}
6
{\displaystyle 6\,}
12
{\displaystyle 12\,}
8
{\displaystyle 8\,}
a Cubo
Tetraedro
2
12
a
3
{\displaystyle {\frac {\sqrt {2}}{12}}a^{3}}
3
a
2
{\displaystyle {\sqrt {3}}a^{2}}
4
{\displaystyle 4\,}
6
{\displaystyle 6\,}
4
{\displaystyle 4\,}
a
Pirámide cuadrada
2
6
a
3
{\displaystyle {\frac {\sqrt {2}}{6}}a^{3}}
(
1
+
3
)
a
2
{\displaystyle (1+{\sqrt {3}})a^{2}}
5
{\displaystyle 5\,}
8
{\displaystyle 8\,}
5
{\displaystyle 5\,}
a
Figuras de superficies curvadas [ editar ]
Nombre
Volumen
Área
Especificaciones
Dibujo
Cilindro
π
r
2
h
{\displaystyle \pi r^{2}h}
2
π
r
h
+
2
π
r
2
{\displaystyle 2\pi rh+2\pi r^{2}}
r h
Cono
π
3
r
2
h
{\displaystyle {\frac {\pi }{3}}r^{2}h}
π
r
r
2
+
h
2
+
π
r
2
{\displaystyle \pi r{\sqrt {r^{2}+h^{2}}}+\pi r^{2}}
r h Cono
Esfera
4
π
3
r
3
{\displaystyle {\frac {4\pi }{3}}r^{3}}
4
π
r
2
{\displaystyle 4\pi r^{2}}
r
Esferoide
4
π
3
a
2
⋅
c
{\displaystyle {\frac {4\pi }{3}}a^{2}\cdot c}
2
π
a
(
a
+
c
e
arcsin
e
)
{\displaystyle 2\pi a(a+{\frac {c}{e}}\arcsin e)}
siendo a c c e excentricidad de la elipse.
Elipsoide
4
π
3
a
b
c
{\displaystyle {\frac {4\pi }{3}}abc}
S
=
2
π
(
c
2
+
b
a
2
−
c
2
E
(
α
,
m
)
+
b
c
2
a
2
−
c
2
F
(
α
,
m
)
)
,
{\displaystyle S=2\pi \left(c^{2}+b{\sqrt {a^{2}-c^{2}}}E(\alpha ,m)+{\frac {bc^{2}}{\sqrt {a^{2}-c^{2}}}}F(\alpha ,m)\right),\,\!}
siendo a b c
Toro
2
π
2
r
2
R
{\displaystyle 2\pi ^{2}r^{2}R}
4
π
2
r
R
{\displaystyle 4\pi ^{2}rR}
r circunferencia rotada), y R
Toroide
2
π
R
A
{\displaystyle 2\pi RA}
2
π
R
P
{\displaystyle 2\pi RP}
A generatriz , y R generatriz .
Figuras de cuatro dimensiones [ editar ]
Nombre
Hiper-volumen
Hiper-área
Poliedros
Caras
Aristas
Vértices
Comentarios
Teseracto
a
4
{\displaystyle a^{4}\,}
8
a
3
{\displaystyle 8a^{3}}
8
{\displaystyle 8\,}
24
{\displaystyle 24\,}
32
{\displaystyle 32\,}
16
{\displaystyle 16\,}
a
Pentácoron
5
96
a
4
{\displaystyle {\frac {\sqrt {5}}{96}}a^{4}}
5
2
12
a
3
{\displaystyle {\frac {5{\sqrt {2}}}{12}}a^{3}}
5
{\displaystyle 5\,}
10
{\displaystyle 10\,}
10
{\displaystyle 10\,}
5
{\displaystyle 5\,}
a
Nombre
Hiper-volumen
Hiper-área
Comentarios
Hiperesfera
π
2
2
r
4
{\displaystyle {\frac {\pi ^{2}}{2}}r^{4}}
2
π
2
r
3
{\displaystyle 2\pi ^{2}r^{3}\,}
r
Figuras de n dimensiones [ editar ]
Familia
Espacio (n)
Espacio (n-1)
Comentarios
Cuadrado, cubo, hipercubo...
a
n
{\displaystyle a^{n}\,}
2
n
a
n
−
1
{\displaystyle 2na^{n-1}\,}
a n
Triángulo equilátero, tetraedro, pentácoron...
n
+
1
n
!
2
n
a
n
{\displaystyle {\frac {\sqrt {n+1}}{n!{\sqrt {2^{n}}}}}a^{n}}
(
n
+
1
)
n
(
n
−
1
)
!
2
n
−
1
a
n
{\displaystyle {\frac {(n+1){\sqrt {n}}}{(n-1)!{\sqrt {2^{n-1}}}}}a^{n}}
a n
Círculo, esfera, hiperesfera...
π
n
2
r
n
Γ
(
n
2
+
1
)
{\displaystyle \pi ^{\frac {n}{2}}r^{n} \over \Gamma ({\frac {n}{2}}+1)}
2
π
n
2
r
n
−
1
Γ
(
n
2
)
{\displaystyle 2\pi ^{\frac {n}{2}}r^{n-1} \over \Gamma ({\frac {n}{2}})}
r n
Spiegel, M. & Abellanas, L.: "Fórmulas y tablas de matemática aplicada ", Ed. McGraw-Hill, 1988, pp. 185-89 ISBN 84-7615-197-7 .
