Mathe - Formeln
 

2 D Flchen

3 D Krper

Das rechtwinklige Dreieck Der Quader
Das gleichseitige Dreieck Der Wrfel
Die Raute Der Zylinder
Das Quadrat Der Kegel
Das Rechteck Die quadratische Pyramide
Das Trapez Der Tetraeder
Das Parallelogramm Der Kegelstumpf
Der Drache Die Kugel
Der Kreis Das dreieckige Prisma
Der Kreisring Das sechseckige Prisma
Der Kreisausschnitt Der quadratische Pyramidenstumpf
Das regelmige Sechseck Der regelm. dreieckige Pyramidenstumpf
  Der regelm. sechseckige Pyramidenstumpf

 

Mathematische Zeichen
Griechische Buchstaben
Maeinheiten
bungsbeispiele: Flchen und Volumsberechnungen

 

 

Mathe - Formeln 2 D Flchen

 

Das rechtwinklige Dreieck

 

a = g - b

a = INV tan a / b

b = g - a

g = a + b

c = a / sina

c = (a + b)

a = c * sina

a = (c - b)

a = b * tana

b = (c- a)

 

 

Das gleichseitige Dreieck

 

u = 3 * a

A = a/ 4 *3

h = a/ 2 *3

a = u / 3

a = ((A * 4) / 3)

a = (h * 2) / 3

a = b = c!!

 

 

Die Raute

 

 

u = 4 * a

A = (e * f) / 2

A = a * h

h = A / a

e = ( 2a - f)

e = (A * 2) / f

a = u / 4

a = ((e / 2) + (f / 2))

a = A / h

 

 

Das Quadrat

u = 4 * a

 

A = a

 

e = a * 2

a = e / 2

a = A

a = u / 4

 

 

Das Rechteck

 

A = a * b

 

u = 2a + 2b

 

e = (a * b)

a = A / b

a = (u - 2b) / 2

a = (e - b)

b = A / a

b = (u - 2a) / 2

b = (e - a)

 

 

Das Trapez

a = u - b - c - d

a = m * 2 - c

b = u - a - c - d

b = u - m * 2 - d

c = u - a - b - d

c = m * 2 - a

d = u - a - b - c

d = u - m * 2 - b

m = (a + c) / 2

m = a / h

m = (u - b - d) / 2

u = a + b + c + d

u = m * 2 + b + d

 

h = a / m

h = a / ((a + c) / 2)

A = m * h

A = ((a + c) / 2) * h

 

 

Das Parallelogramm

 

A = a * h

A = b * h

u = 2 * (a + b )

h= A / b

h= A / a

b = U / 2 - a

b = A / h

a = U / 2 - b

a = A / h

  

Der Drache

a = (u / 2) -b

b = (u / 2) - a

e = (A * 2) / f

A = (e * f) / 2

f = (A * 2) / e

f =a - (e / 2) + b - (e / 2)

 

 

Der Kreis

 

 

u = 2 * p * r  A = p * r  d = 2 * r r = d / 2

r = (A / p)

r = u / (2 * p)

 

 

Der Kreisring

 

  

A = p * (r - r)

u = 2 * p * (r- r)

r= ((A / p) + r)

r= u / ( 2 * p) - r

r= ((A / p) + r)

r= U / ( 2 * p) - r

 

 

Der Kreisausschnitt

 

  

a = (A * 360) / (p * r)

A = (b * r) / 2

b = (A * 2) / r

b = 2 * p * r * a / 360

r = (A * 2) / b

r =(A * 360) / (p * a)

r = (b * 360) / (a * 2 * p)

 

 

Das regelmige Sechseck

  

  

u = 6 * a

A = 3 / 2 * a * 3

A = 6 * (a / 4 * 3)

h = a / 2 * 3

a = h * 2 / 3

a = ((a / 6) * 4) / 3

a = u / 6

 

 

 

Mathe - Formeln 3D Krper

 

Der Quader

 

 

O = 2 (a * b + b * c + a * c)

V = a * b * c

b = ((O / 2) - a * c) / ( a + c)

b = V / (c * a)

b = (e - c - a)

c = ((O / 2) - a * b) / ( a + b)

c = V / (b * a)

c = (e - a - b)

e = (a + b + c)

a = ((O / 2) - b * c) / ( b + c)

a = V / (c * b)

a = (e - b - c)

 

 

Der Wrfel

 

4

V = a

e = a * 3

a = (O / 6)

a = e / 3

a = V = V hoch 1/3

 

 

Der Zylinder

 

 

A = r * p

A = (O - M) / 2

V = p * r * h

O = 2 * p * r *(r + h)

O = M + 2 * A

M = O - (2 * A)

r = (V / (h * p))

r = (A / p)

r = (( O - M) / (2 * p))

h = (O / (2 * r * p)) - r

h = V / (r * p)

 

 

Der Kegel

 

 

a = (r / s) * 360

h = (s - r)

h = (s - r)

h = V / ((p / 3) * r)

O = p * r + p * r * s

G = O - M

G = p * r

s = M / (p * r)

s = (r * 360) / a

s = (r + h)

s = (r / a) * 360

r = (G / p)

r = V / (h * (p / 3))

r = (a / s) / 360

r = (s - h)

r = -(s / 2) + ((s / 2) + (O / p))

V= (p / 3) * r * h

 

 

Die quadratische Pyramide

  

  

a = ((3*V) / h)

a = 2 (hs - h)

a = 2 (s - hs)

a =(M / hs) / 2

a =G

a =(O - M)

h = (V * 43) / a

h = (hs - (a / 2))

hs = M / ( 2 * a )

hs = (h + ( a / 2 ))

hs = (s - (a / 2 ))

hs = (O - a) / 2a

O =a + 2a * hs

O = G + M

G = a

G = O - M

s = (hs +( a / 2))

V= (1/3) * a * h

V = (1 / 3) * G * h

 

 

a = Grundkante; h = Hhe ber dem Mittelpunkt; hs = Seitenhhe; O = Oberflche

M= Mantel ( 4 Dreieck); G = Grundflche; V = Volumen; s = Seitenkantenlnge;

 

Der Tetraeder

 

 

O = a * 3

V = (a / 12) * 2

a = (O / 3)

a = ( (V * 12) / 2)

d = u - a - b - c

d = u - m * 2 - b

 

 

Der Kegelstumpf

 

 

s = (h + ( r1 - r2))

s = M /(p *( r1 +r2))

A1 = p * r 1

A2 = p * r2

O = p * (r1+r2) + p * s * (r1 + r2)

O = M + A1 +A2

M = p * s *(r1 + r2)

r1 = ((s - h)) + r2

r1 = (M / ( p * s)) - r2

r1 = (A1 / p)

h = (V * 3) / ( p *(r1 + r1 * r2 + r2)

h = (s - (r1 - r2))

 

 

Die Kugel

 

 

O = 4 * p * r

V = (4 / 3) * p * r

d = 2 * r

r = d / 2

r = (O / (4 * p))

r = ( V / (4 / 3 * p))

 

 

Das dreieckige Prisma

 

M = U * h

M = 3 * a * h

M = O - 2 * A

O = M + 2 * A

O = 3ah + (2 * a * 3) / 4

O = a ( 3h + ((a3) / 2))

h = ( V * 4) / (a * 3)

h = M / U

h = M / ( 3 * a)

h = V / A

h = (( O / a) - ((a3) / 2) ) / 3

a = U / 3

a = (2 * hd) / 3

a = (4 * (A / 3))

a = (( 4 * V) / (3 * h))

a = M / ( 3* h)

U = 3 * a

hd = (a / 2) * 3

A = ( a / 4) * 3

A = ( O - M) / 2

A = V / h

V = A * h

V = (( a * h) / 4) * 3

 

 

Das sechseckige Prisma

 

 

hd = (a / 2) *3

U = 6 * a

h = V / A

h = M / U

h = M / ( 6 * a)

h = ( V * 2) / ( 3 * 3a)

h = ((O / (3a)) - a3) / 2

a = ( hd *2) / 3

a = (((A / 6) * 4) / 3)

a = U / 6

a = M / ( 6 * h)

a = ( ( V * 2) / (3 * 3 * h) )

V = A * h

V = ((3a * h) / 2) * 3

M = 6 * a * h

M = U * h

M = O - 2A

 

A = (3a / 2) * 3

A = 6 * (a / 4) * 3

A = V / h

A = O - M / 2

O = M + 2 * A

O = 6 * a * h + ((2 * 3a * 3) / 2)

O = 3 * a * (2h + a3)

 

 

Der quadratische Pyramidenstumpf

 

 

h = (V / ( a1 + a1 * a2 + a2)) * 3

h = (s - ((a1 - a2) / 2))

h = (hs - ((a1 - a2) / 4))

a2 = A2

a2 = ((2*(s - h)) - a1) * -1

a2 = ((4 * (s - hs)) - a1) * -1

a2 = ((4 * (hs - h)) - a1) * -1

a2 = M / (2 * hs) - a1

M = O - A1 - A2

M = 2 * hs * (a1 + a2)

V = ( h / 3) * (a1 + a1 * a2 + a2)

s = (h + ((a1 - a2) / 2))

s = (hs + ((a1 - a2) / 4))

hs = (h +((a1 - a2) / 4))

hs = (M / (a1 + a2)) / 2

hs = (s - ((a1 -a2) / 4 ))

hs = (( O - a1 - a2) / (a1 + a2)) / 2

a1 = (4 * (hs - h)) + a2

a1 = A1

a1 = (2 * (s - h)) + a2

a1 = ( 4 * (s - hs)) + a2

a1 = M / (2 * hs) - a2

O = a1 + a2 + 2 * hs * (a1 + a2)

O = M + A1 + A2

A1 = a1

A1 = O - M - A2

 

 

A2 = a2

A2 = O - M - A1

 

 

Der regelm. dreieckige Pyramidenstumpf

 

 

h = ((V / ( a1 + a1 * a2 + a2)) * 12 / 3)

h = (s - ((a1 - a2) / 3))

h = (hs - ((a1 - a2) / 12))

a2 = (A2 * 4 / 3)

a2 = ((3*(s - h)) - a1) * -1

a2 = ((4 * (s - hs)) - a1) * -1

a2 = ((12 * (hs - h)) - a1) * -1

a2 = (2 * M) / (3 * hs) - a1

V = ( h * 3 / 12) * (a1 + a1 * a2 + a2)

s = (h + ((a1 - a2) / 3))

s = (hs + ((a1 - a2) / 4))

O = ((a1 + a2)* 3) / 4 + (3 * hs * (a1 + a2) / 2)

O = M + A1 + A2

a1 = (12 * (hs - h)) + a2

a1 = (A1 * 4 / 3)

a1 = (3 * (s - h)) + a2

a1 = ( 4 * (s - hs)) + a2

a1 = (2 * M) / (3 * hs) - a2

hs = (h +((a1 - a2) / 12))

hs = ((2 * M) / (a1 + a2)) / 3

hs = (s - ((a1 -a2) / 4 ))

hs = (( O - (((a1 + a2) * 3) / 4)) / ((a1 + a2) / 2)) / 3

 

 

 

Der regelm. sechseckige Pyramidenstumpf

 

 

h =(( V / (a1 + a1 * a2 + a2)) * 2 / 3)

h = (s - (a1 - a2))

h = (hs - (( 3 * (a1 - a2)) / 4))

a2 = (A2 / ( 3 / 2 * 3))

a2 = M / (3 *hs) - a1

a2 = ((s - h) - a1) * -1

a2 = (( 4 * (s - hs)) - a1) * -1

a2 = (((hs - h) * 4 / 3) - a1) * -1

V = (( h * 3) / 2) * (a1 + a1 * a2 + a2)

 

s = (h + (a1 - a2))

s = (hs + ((a1 - a2) / 4))

O = ((3 * 3) / 2) * (a1 + a2) + 3 * hs * (a1 + a2)

O = M + A1 + A2

a1 = (A1 / ( 3 / 2 * 3))

a1 = M / (3 *hs) - a2

a1 = (s - h) + a2

a1 = ( 4 * (s - hs)) + a2

a1 = (((hs - h) * 4 / 3)) + a2

hs = (h + ((3 * (a1 - a2)) / 4))

hs = (s - ((a1 -a2) / 4 ))

hs = (M / 3) / ( a1 + a2)

hs = ( O -3 * (3) / 2 * (a1 + a2)) / 3 / (a1 + a2)

 

 

 

 

Mathematische Zeichen

 

=

gleich

Winkel

<

kleiner als

rechter Winkel(90)

>

grer als

Natrliche Zahlen ohne Null

<

kleiner oder gleich

Ganze Zahlen

>

grer oder gleich

Natrliche Zahlen

ungefhr gleich, rund

AB

Gerade durch A u. B

 
=

entspricht

__
AB

Strecke zwischen A u. B

unendlich

Rationale Zahlen

||

parallel

Reelle Zahlen

^

rechtwinklig, senkrecht zu

@

kongurent

~

proportional, hnlich

ungleich

 

 

 

 

 

  

Griechische Buchstaben

 

a

Alpha

o

Omikron

b

Beta

p

Pi

g

Gamma

t

Tau

d

Delta

u

Ypsilon

e

Epsilon

w

Omega

z

Zeta

x

Xi

h

Eta

y

Psi

l

Lambda

m

My

i

Jota

j

Phi

 

 

Maeinheiten

 

Zeitmae =

1 Jahr = 365 Tage
               1 Tag = 24 h
                            1 h = 60 min
                                      1 min = 60 s

Gewichtsmae =

1 Mt = 1000 kt
                 1 kt = 1000 t
                                 1 t = 1000 kg
                                               1 kg = 1000 g
                                                              1 g = 1000 mg

Lngenmae =

1 km = 1000 m
                1 m = 10 dm
                           1 dm = 10 cm
                                       1 cm = 10 mm 

Flchenmae =

1 km = 100 ha
                 1 ha = 100 a
                               1 a = 100 m
                                         1 m = 100 dm 
                                                        1 dm = 100 cm
                                                                       1 cm = 100 mm

Raummae =

1 m = 1000 dm
                  1 dm = 1000 cm
                                    1 cm = 1000 mm 

Hohlmae =

1 hl = 100 l
           1 l = 1000 ml                       1 dm = 1 l
                      1 ml = 1  cm 

 

 

bungsbeispiele - Volumsberechnungen

 

Volum1.jpg Volum2.jpg Volum3.jpg
     
Volum4.jpg Volum5.jpg  

 Copyright 2005 Hojager, Gerhard J. Hojas, Linz/Ebelsberg. Alle Rechte vorbehalten.