In this work I investigated different shaped tubes, which can be made from rectangular sheets of card, of given width and length. I used Microsoft Excel to collate data.

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          To find the volumes of the different shaped tubes, I worked out the cross sectional area and then multiplied that by the length of the tube.         General Formulas for volumes of prisms                  Cylinder: cross sectional area = πr2                          Volume = πr2 x length         Cuboid with rectangular end face: cross sectional area = width x length                              Volume = cross sectional area x length of prism         Equilateral and Isosceles Triangle face: cross sectional area = 0.5 x base x height                                                  Volume = cross sectional area x length of prism         Hexagon: cross sectional area =                   Generalised Formula for working out the volume of any polygon Results: Refer to appendix 1 for the data collected for the volumes of a cylindrical tube.         Refer to appendix 2 for the data collected for the volumes of a rectangular tube.         Refer to appendix 3 for the data collected for the volumes of an equilateral triangular tube.         Refer to appendix 4 for the data collected for the volumes of an isosceles triangular tube.          This table shows the largest possible volumes possible with 24cm as the length and 32cm as the length.                    Length 24cm and Width 32cm ShapeVolume (cm3)Circle1995.696Rectangle1536Equilateral Triangle1182.413Isosceles Triangle1175.755Hexagon1773.620027            Length 32cm and Width 24cm ShapeVolume (cm3)Circle1466.772Rectangle1152Equilateral Triangle886.81Isosceles Triangle886.81Hexagon1330.21502 Conclusions: Using the results above.  I concluded that: A cylinder has the largest volume.  You always get a larger volume when the perimeter of the cross section is 32cm and the length is 24cm.  This is illustrated with the larger volume of all the shapes when the lengths of the tubes are 24cm and not 32cm. As the number of sides of the shape begins to increase, so then does the volume of the tube in both cases. Part 2 Introduction: This part of the investigation will involve me using a piece of card with a fixed area; to investigate the volumes of open ended tubes that I can obtain from it.  I have used Microsoft Excel once again to help me to obtain my results.   Method: Using the knowledge I gained from the previous part of the investigation.  I decided to investigate the sizes of cylindrical tubes, which could be made from this card.  The card has a fixed area of 1200cm2, this is how we worked out the volume of the possible cylindrical tubes that we could get from it. Results: Refer to appendix 5 for the results of volumes of the cylindrical tube         Also refer to graph 1 Conclusions: Using the results and graph above, I have concluded that The cylinder