10.7cm
Hexagon
Cylinder
Paper 32cm by 24cm
Second method: using the paper portrait
Cube
Triangular Prism
Hexagon
Cylinder
All the results presented in the table below:
Conclusion for Investigation No.1
To conclude investigation 1, for any tubes constructed with either using the paper landscape or portrait, the biggest volume found is when the tube has more sides but eventually, after i.e. when the tube have infinity sides then the tube will then become a cylinder so the biggest volume found is when it is a cylinder. And also the tubes constructed with the paper landscape are always bigger in volume than the tubes constructed with the paper portrait.
Investigation No.2
The aim for investigation no.2 is to investigate the volume of open-ended tubes, which can be made from any rectangular piece of card. Method no.1 is used for this investigation because I found out those short and wide tubes holds a larger volume than the high and thin tubes in investigation no.1.
Take an open-ended cube for example:
If the piece of rectangular paper is 24cm by 32cm, to construct a cube by using the paper landscape then the height will be 24cm with the length and width 8cm. The volume of this cube is 1536 cm3. But what will happen if the 24cm by 32cm paper was cut into half horizontally, given the cube’s height 12cm with the length and the width 16cm? And what will also happen if I cut the 24cm by 32cm paper in half horizontally two times given the cube’s height 8cm with the length and width 24cm and then what will happen if I cut the 24cm by 32cm paper three times and so on. What effect does it have on the volume if the height of the cube is lower with the width and length longer?
The diagrams below will help you to have a better understanding of this investigation:
In the table below are some volumes that I found when I cut the 24cm by 32cm rectangular paper with numbers of times.
Conclusion for Investigation No.2
From the results that I observed in investigation 2 the conclusion is very simple, the lower the height of the tube is, the bigger the volume will be but the area of the paper is always the same.
Investigation No.3
The aim in investigation 3 is for a given area of a rectangular piece of card, investigate the volumes of open-ended tubes, which can be made. There are two sections in this investigation. The first part is to investigate a formula that can be use to calculate the volumes of open-ended polygon tubes for any given area. The second part is to investigate a formula that can be use to calculate the volumes of open-ended cylinder tubes for any given area.
Formula for polygon:
Take an open-ended hexagon tube for example:
Hexagon
1.
2.
3.
4.
5.
6.
7.
To in order to find the formula that work for every polygon I will substitute the letters in instead of the numbers. And I will follow the steps in the box above.
Explanation of the formula:
The first equation explains 1 – 3 steps in the first box on this page. The ‘h’ is the height of triangle, ‘w’ is the width of the paper and ‘n’ is the number of side in a polygon. The width on the paper divided by number of sides in a polygon can find the length of each side. And divided the length of one side by 2 then you can find the length of half a side. The right angle triangle is formed therefore trigonometry can be use. The next step is to find the angle inside the polygon. 360 degrees is always the sum of all polygons. So you divided 360 by the number sides, which is ‘n’, then you will get the angle of the top angle.
We know the triangle is an isosceles triangle so the two bottom angles are the same and 180 degrees is the total sum for a triangle thus 180 – the top angle then divide by two then you will get the bottom two angles and the bottom two angles should be identical. The following step is to multiply tangent the top angle by the length of half a side. After this, you can find the height of the triangle.
Then multiply the height of triangle to the length of side then you will find the area of the square. Then divide it by 2 to get the area of the triangle. Then multiply the area of triangle to the ‘n’ side to find the total area of the polygon. Afterward, multiply the area of polygon to the height of tube to find the whole volume of the polygon tube.
Here is the formula that can calculate the volume of polygon tube for any given area of a rectangular piece of paper
But however this formula can be simplify:
Formula of Cylinder
The area of circle is =
The circumference is =
But buried in mind, the circumference of the circle is the width of the paper so
can be written as = w
