# volumes of open ended prisms

Extracts from this document...

Introduction

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

Middle

Area of cross section = 3 x 9 = 27cm2

Volume of prism = 27 x 32 = 864cm3

3cm

32cm

9cm

Area of cross section = 2 x 10 = 20cm2

32cm Volume of prism = 20 x 32 = 640cm3

2cm

10cm

I have also worked out a formula to work out the volumes of quadrilateral:

c V= length of prism. Multiplies

with area to gives us the volume.

a

This gives us the area of the cross section

b

To show that this formula will give me the volume of a quadrilateral, I must put the figures for one of my quadrilaterals in and check to see whether or not the formula gives me the correct volume.

A = 6cm, B= 6cm, C = 32cm

6 x 6 x 32 = 1152cm3

1152cm3 isthe same volume as I previously worked out, and therefore proves that this formula is correct.

I have realised that the square is the quadrilateral give the largest volume. Taking into account both the results I have obtained from the triangular and quadrilateral prisms, I conclude that regular shapes give larger volumes than irregular shapes do, therefore I will now only work out the volumes for various regular shapes.

To prove that infact the square will give us the largest volume; I will consider values close to the dimensions of the square.

Value no.1 | Value no.2 | Value no.3 | Value no.4 | |

A | 5.9 | 5.8 | 5.7 | 5.6 |

B | 6.1 | 6.2 | 6.3 | 6.4 |

Volume | 1151.68cm3 | 1150.72cm3 | 1149.12cm3 | 1146.883 |

32cm

A

B

Conclusion

Because we know that regular shapes give us the largest volume I will only work out the volumes for regular prisms.

### Squares

I will work out the volumes for the various squared, by using the formula I worked out in part 1:

AB x C

c

a

b

Length of prism (cm) | Perimeter (cm) | Length of sides (cm) | Volume (cm3). |

1 | 1200 | 300 | 90000 |

2 | 600 | 150 | 45000 |

3 | 400 | 100 | 30000 |

4 | 300 | 75 | 22500 |

5 | 240 | 60 | 18000 |

6 | 200 | 50 | 15000 |

### Equilateral triangle.

I will also use the formula I previously formed to work out the volume of the triangular:

y

a c

b

c2 - b2 x b x y

2 _

2

Length of prism (cm) | Perimeter of triangle (cm) | Length of sides (cm) | Volume (cm3). |

1 | 1200 | 400 | 69282.0323 |

2 | 600 | 200 | 43641.01615 |

3 | 400 | 133.3 | 26628.32952 |

4 | 300 | 100 | 17320.50808 |

5 | 240 | 80 | 13856.40646 |

6 | 200 | 66.6 | 11523.92292 |

#### Icosagon

Out of the polygons I investigated, the 20 sided regular icosagon had the largest volume, therefore I will now work out the volume for an icosagon using the formula for an ‘n’ sided shape:

P/2N

Tan180 x PL

## N

2

Length of prism (cm) | Perimeter of icosagon (cm) | Volume (cm3). |

1 | 1200 | 113647.5273 |

2 | 600 | 56823.76363 |

3 | 400 | 37882.50909 |

4 | 300 | 28411.88182 |

5 | 240 | 22729.50545 |

6 | 200 | 18941.25454 |

From this whole investigation, I have found out that a cylinder will give us the largest volume, this may be due to the fact that it is has a smooth curve rather than many sides. Now, I understand that as the Perimeter of a prism increases and the length decreases the volume increases. So, if I were to make a container that would be able to carry the largest amount of water, I would take into account the fact that cylinders with a large perimeter yet small length will give the largest volume.

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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