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formulae

VIEWS: 17 PAGES: 7

									                                                                           T E C H N I C A L D ATA S H E E T




                formulae                                                           (equilateral triangle, pentagon, hexagon,
                                                                                   octagon, polygon)


                                                                                                                     equilateral triangle
                                     2
                                 a           3
                  A      =       4




                                                                                                                                            a
                                 a


                                                                                               h
                  h      =                   3
                                 2

                                                                                                                                    a
                                 5
                  A      =               r2       10           + 2           5                                               pentagon
                                 8
                                                                                                                                C
                                 1
                  a      =                r       10           -       2     5
                                 2
                                 1                                                                                     B                        E
                  !      =                r        6           +       2     5
                              4                                                                                  A        !             D
                  Construction:
                                                                                                            a




                                                                                                                                    r
                  AB     =      0.5      r,       BC       =       BD ,       CD = CE

                                 3                                                                                           hexagon
                  A      =                a2      3
                                 2
                  d      =       2       a
                                                                                                                                        s



                                     2        s                              s
                         =                             =
                                                                                              d




                                                                   1.155
                                                                                                        a




                                     3
                                     3
                  s      =                    d        =           0.866     d
                                     2
                                 3
                  A      =               a s   = 0.83 s2                                                                     octogan
                                 2
                         =       2       s d 2 = s2
                  a      =       s    x tan           22.5° ~
                                                            ~              0.451   s
                                                                                                                                        d
                                                                                                        a
                                                                                              s




                  s      =       d x cos              22.5° ~
                                                            ~              0.942   d
                                        s
                  d      =               cos 22.5°
                                                                   ~
                                                                   ~       1.083   s

                                                                                                                              polygon

                  A      =       A1          +    A2       +       A3
                                                                                                                         b               A3
                                                                                                    h1
                                                                                                                                                             3
                                                                                                                                                        h




                                 ah1         +    bh2      +       bh3                                A1           A2
                         =
                                                       2
                                                                                                                                2
                                                                                                                           h




                                                                                                                                                                 Revision: 01-05-08
                                                                                 a.b.e. is an ISO 9001 : 2000 Registered Company.




a.b.e. Construction Chemicals Limited                                  P.O. Box 23053, Isipingo, 4110                        7 Wilcox Road, Isipingo, 4133              Tel. (031) 913 5400
                                                                  T E C H N I C A L D ATA S H E E T




                formulae                                                     (circle, annulus, sector of a circle,
                                                                             segment of a circle, ellipse)

                                   " d2                         " r2
                                                                                                                             circle
                    A       =      4                  =

                            ~
                            ~       0.785             d2

                                         " r                    " d




                                                                                                                                    r
                                                                                                                        d
                    U       =       2                  =


                                                                                                                            annulus
                                   "      (D2 =                d 2)
                    A       =      4

                            =       " (d          +        b) b                                                     b       d
                                   D - d
                    b       =
                                         2

                                                                                                                                D
                                   "             r2 r                        a r2                                  sector of a circle
                    A       =    360°
                                                                   =         2
                                  br
                            =       2                                                                                           !
                                                                                                                                        r


                                   "             ra
                    b       =     180°
                                   "
                     a      =     180°
                                             a         (a       = a in circular measure)
                                                                                                                                b
                                                           a                                                      segment of a circle
                     s      =      2     r s     in
                                                           2                      2
                                    h                       r                                                                   !            r
                    A       =
                                    6s
                                       ( 3 h2 + 4 s2 ) = 2 ( a - sina)
                                    h       2
                                                                                                                        s
                    r       =          + 3
                                    2     8h
                                               a        s          a
                    h       =      r ( 1 - cos   ) = 2 tan 4
                                                                                                                                                  h




                                               2
                                   "
                     a      =     180°
                                         a ( a = a in circular measure )                                                        b
                                   "
                    A       =            Dd           = " a b                                                               ellipse
                                   2                                                                                a
                                         D + d
                                   "
                                                                             2
                    U       ~
                            ~
                                             2
                                                                         d       0.785
                                                                                                                                        b




                                                                  1              1           1
                            =      " (a + b)              1 +           #2 +          #4 + 256 #6    +
                                                                                                                                                  d




                                                                  4              64

                                    25                                                a-b
                                  16384
                                                      #8 + …           , where    # = a+b
                                                                                                                                D
                                                                                                                                                      Revision: 01-05-08
                                                                         a.b.e. is an ISO 9001 : 2000 Registered Company.




a.b.e. Construction Chemicals Limited                           P.O. Box 23053, Isipingo, 4110                    7 Wilcox Road, Isipingo, 4133              Tel. (031) 913 5400
                                                         T E C H N I C A L D ATA S H E E T




                formulae                                          (square, rectangle, parallelogram,
                                                                  trapezium, triangle)

                                                                                                           square

                A       =      a2

                c       =       A




                                                                                        a




                                                                                                              d
                d       = a        2



                                                                                                         rectangle


                A       =     a b
                                                                                                            d
                                                                                        b


                d       =        a2 + b2
                                                                                                                a

                                                                                                       parallelogram

                A       =     a b       = a b   sin!

                                                                                                                    d1
                                    ( a + h cot !) 2 + h
                                                         2
                d1      =
                                                                                        h




                                                                                                                         d2
                                                                                              !
                                    ( a - h cot !) 2 + h
                                                         2
                d2      =
                                                                                                          a


                                                                                                         trapezium
                            a + b                                                                                   a
                A       =               h = m h
                               2


                            a + b
                                                                                        h




                m       =                                                                                       m
                               2

                                                                                                                b

                              ah                                                                           triangle
                A       =                  = " s
                               2


                        =       s ( s - a) (s - b) (s - c)
                                                                                                  c




                                                                                                                              b
                                                                                  h




                            a+b+c                                                                           "
                s       =
                                2

                                                                                                              a

                                                                                                                                         Revision: 01-05-08
                                                                a.b.e. is an ISO 9001 : 2000 Registered Company.




a.b.e. Construction Chemicals Limited                  P.O. Box 23053, Isipingo, 4110                    7 Wilcox Road, Isipingo, 4133          Tel. (031) 913 5400
                                                            T E C H N I C A L D ATA S H E E T




                formulae                                             (cylinder, hollow cylinder, cone,
                                                                     frustum of cone, sphere)

                               !                                                                                cylinder
                V       =               d2 h
                                4
                                                                                                                               r
                Am =                ! r h




                                                                                           h
                               2



                 A0 =          2    ! r (r + h)
                                                                                                           d

                                                                                                          hollow cylinder


                               !
                V       =              h (D 2 - d 2 )
                                                                                           h

                                4                                                                                      d


                                                                                                          D

                               !                                                                                   cone
                V       =                r2h
                               3                                                                                                       A2
                Vm = ! r m
                                                                                               x
                                                                                           h




                 A0 = ! r (r + m)
                                                                                                                                   m




                m      =       h2 + r2
                A2 : = A1 = x 2 h 2                                                                   r                                A1

                               !
                V       =                h ( D 2 + Dd + d 2 )                                             frustum of cone
                              12
                                                                                                                                   m




                               ! m(D+d)=2!ph
                                                                                                                                   m
                                                                                                                                   2




                Am =
                                                                                           h




                               2                                                                               d

                                       D-d                                                                                 p
                m      =           (     2
                                               )2 + h 2
                                                                                                          D
                               4                    1
                V       =                ! r2 =            !d2                                                   sphere
                               3                    6

                        ~     4 . 189     r3
                                                                                           d




                        ~

                                                                                                                           r
                 A0 =         4     ! r3 = ! d 2



                                                                                                                                               Revision: 01-05-08
                                                                   a.b.e. is an ISO 9001 : 2000 Registered Company.




a.b.e. Construction Chemicals Limited                     P.O. Box 23053, Isipingo, 4110                       7 Wilcox Road, Isipingo, 4133          Tel. (031) 913 5400
                                                          T E C H N I C A L D ATA S H E E T




                formulae                                              (cube, cuboid, parallelepiped,
                                                                      pyramid, frustum of pyramid)

                                                                                                              cube
                V       =     a2 h

                                    2
                A0 =           6a




                                                                                     a




                                                                                                                d
                d       =               3   a                                                                                    a
                                                                                                              a

                                                                                                            cuboid

                V       =    a b c

                A0 =         (a b + a c + b c)
                                                                                     a




                                                                                                                d
                 d      =        a2 + b2 + c2




                                                                                                                                a
                                                                                                              a

                                                                                                        parallelepiped



                V       =    A1 h
                                                                                     h




                (Cavalieri principle)
                                                                                                                                 A1


                                                                                                            pyramid
                                                                                     h




                              A1 h
                V       =
                                 3



                                                                                                                                 A1

                                                                                                     frustum of pyramid

                                                                                                                                 A2
                                  h             2   2
                 V      =                   ( A +A +         A1 A2 )
                                  3
                                                                                     h




                        ~
                        ~         h A1 + A2
                                                2
                                                                                                                                 A1


                                                                                                                                          Revision: 01-05-08
                                                                 a.b.e. is an ISO 9001 : 2000 Registered Company.




a.b.e. Construction Chemicals Limited                   P.O. Box 23053, Isipingo, 4110                    7 Wilcox Road, Isipingo, 4133          Tel. (031) 913 5400
                                                                   T E C H N I C A L D ATA S H E E T




                formulae                                                    (zone of sphere, segment of sphere, sector of sphere,
                                                                            sphere with cylindrical bore, sphere with conical bore)

                                                                                                                  zone of a sphere
                                 !                                                                                            a
                V      =
                                 6
                                      h   ( 3a 2+ 3b 2+ h 2)




                                                                                              h
                Am =         2       ! r h

                A0 =          ! ( 2 r h + a2 + b 2 )                                                     b                        r

                                 !                3                                                              segment of a sphere
                V      =
                                 6
                                      h   (       4
                                                      s2 + h2 )

                       =     ! h (r- h )
                                                                                              h

                                                  3

                Am =         2       ! r h

                                 !                                                                                                r
                       =              ( s2 +      4h
                                                       2
                                                           )                                                       s
                                 4

                                                                                                                  sector of a sphere
                             2
                V =                   ! r h   2
                                                                                                                                         r
                             3

                              ! r(4h+ s)
                A0 =
                                                                                              h




                                2

                                                                                                             s

                                                                                                       sphere with cylindrical boring
                             !
                V =                  h3
                             6
                                                                                              h




                A0 =        2        ! h (R+r)                                                                                    R


                                                                                                                   r

                                                                                                         sphere with conical boring
                             2
                V =                  ! r h    3
                             3
                                                                                              h




                                                                      2
                A0 =        2        ! h (h +                  r2 = h                                                             r
                                                                      4




                                                                                                                                                       Revision: 01-05-08
                                                                          a.b.e. is an ISO 9001 : 2000 Registered Company.




a.b.e. Construction Chemicals Limited                            P.O. Box 23053, Isipingo, 4110                        7 Wilcox Road, Isipingo, 4133          Tel. (031) 913 5400
                                                           T E C H N I C A L D ATA S H E E T




                formulae                                           (torus, sliced cylinder, ungula, barrel, prismoid)

                                                                                                                  torus

                               !     2
                V      =
                                 4
                                         Dd2




                                                                                           d
                A0 =           !2 D d
                                                                                                                   D


                                                                                                          sliced cylinder


                               !
                V      =
                                 4
                                         d2h
                                                                                          d




                                                                                                          d
                                                                                                                  ungula
                               2
                V      =
                                 3
                                         r2h

                Am =         2     r h
                                                                                          d




                A0 =         A 0 + ! r 2 + ! r r2 + h 2
                                            2        2
                                                                                                      d
                                                                                                                  barrel


                              !
                V      =
                              12
                                         h ( 2 D 2+ d 2)
                                                                                          h




                                                                                                               D


                                                                                                              d
                                                                                                               prismoid
                              h
                V      =                 ( A1   +A2+ 4A)
                                 6                                                                                                  A2
                This formula may be used for calculations                                                                           A
                                                                                          h




                involving solids shown on formulae cube and
                thus speres and parts of spheres.
                                                                                              h
                                                                                                  2




                                                                                                                                    A1

                                                                                                                                              Revision: 01-05-08
                                                                  a.b.e. is an ISO 9001 : 2000 Registered Company.




a.b.e. Construction Chemicals Limited                    P.O. Box 23053, Isipingo, 4110                       7 Wilcox Road, Isipingo, 4133          Tel. (031) 913 5400

								
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