volumes of open ended prisms

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Part 1

For part 1 of this piece of math coursework I will be investigating volumes of prisms, which can be made from a 24cm, by 32cm piece of card. I will be trying to determine which shape will make the prism with the largest volume. To do this, I will be exploring the volumes of triangular prism, cylinders, quadrilateral prisms, pentagonal prism, hexagonal prism, heptagonal prism and octagonal prism. I will then try to work out a formula for working out the volume of an “n” sided shape.  

Triangular prisms

        First, I will be investigating the volume of triangular prisms. We know that in a triangle, the lengths of the left and right sides must add up to more than the length of the base and we also know that the volume of any prism is the area of cross section multiplied by the length.

                                               To find the volume, we must first find the area of the

7cm                                         cross-section. To find the area of a triangle we must                                                    

          h                                       use the formula Area(a) = base (b) x height (h)                                                                                                           

                                                                                                       2

   10cm                32cm                    To work out the height we must use                                              

                                                        Pythagoras’s theorem

 

                           Using the rules of Pythagoras, we know that the height2 = 72-52

         H     7cm    therefore, the height is 4.899cm2

                           So to find the area we do: 4.899 x 10 (base)

            5cm                                                             2                    = 24.49cm 3

           

To work out the volume we must do: area of cross section x length

Therefore the volume is, 24.49 x 32 = 783.84cm3 

                   32cm                   The first step I will take to finding the volume of this

                                               prism will be to find the length of the height. I will

8cm          8cm                       use Pythagoras again and say that the height 2 = 82  - 42      

             h                                so, the height is 6.928cm, to find the area we must

                                               Multiply the height by the base:

        8cm                                6.928 x 8 = 55.426cm 

  1. = 27.713cm2

Now to find the volume: 27.713 x 32 = 886.816cm3

                   32cm

                                                H2 = 92 - 32    h= 8.49cm.  Area = base (6) x height (8.49)   

9cm           9cm                                                                                            2

         H                                                                                       = 25.455cm2

          6cm      

  Volume = area of cross section (25.455) x length of prism (32) → 814.56cm3 

   

         

                32cm                  area of cross section = b x h

                                          h2 = 102 – 42.    h= 9.798cm.  

10cm                                 area = 9.798 x 4 = 19.595cm2

          h       10cm                                2        

         4cm                           Volume= 19.595 x 32 = 627.072cm3 

Out of the triangular prisms, I have realised that the equilateral triangle has given me the largest volume.

        I have also worked out a formula for both the equilateral and isosceles triangle:         y

         

 

 a                c                                we multiply the b and h & divide by 2 to get the                                         area of cross section

           b                

                                   

Volume

Here we are using Pythagoras’                We then multiply by the length

To work out the height.                                of the prism to get the volume.

                                         

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        To prove that this formula works I will input values for the equilateral triangle and confirm that this formula will give me the volume:

A = 8cm, B = 8cm, C = 8cm, Y = 32cm.

82 -   8   2    x  8  x  32                        1773.62

        2                                →              2             →   886.61cm3

           2

The fact that the formula has given me the correct answer, which I worked out previously, proves that this formula gives us accurate volumes for equilateral and isosceles triangular prisms.

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