I am going to begin by investigating a square with a side length of 12 cm. Using this side length, the maximum whole number I can cut off each corner is 5cm, as otherwise I would not have any box left.
I am going to begin by looking into whole numbers being cut out of the box corners.
The formula that needs to be used to get the volume of a box is:
Volume = Length * Width * Height
If I am to use a square of side length 12cm, then I can calculate the side lengths minus the cut out squares using the following equation.
Volume = Length – (2 * Cut Out) * Width – (2 * Cut Out) * Height
Using a square, both the length & the width are equal. I am using a length/width of 12cm. I am going to call the cut out “x.” Therefore the equation can be changed to:
Volume = 12 – (2x) * 10 * x
If I were using a cut out of length 1cm, the equation for this would be as follows:
Volume = 12 – (2 * 1) * 12 - *(2 * 1) * 1
So we can work out through this method that the volume of a box with corners of 1cm² cut out would be:
(12 – 2) * (12 – 2) * 1
10 * 10 * 1
= 100cm³
I used these formulae to construct a spreadsheet, which would allow me to quickly and accurately calculate the volume of the box. On the next page are the results I got through this spreadsheet.
In each previous case, the length of the cut-out corners has been a 1/6 of the size of the square pieces of card. I predict that for a square size piece of card 30x30, the side of the small square cut-outs will be 5cm.
Turn to next 2 pages. >
From the tables and graphs I can see that the square cut-out of 5cm gave the largest volume of the open box. I can see that my prediction was correct so therefore I have now worked out a formula for square shaped pieces of card.
Formula to give the maximum open box volume for square shaped pieces of card:
X= Square sized piece of card
X/6= Side of cut-out corner
Volume= Length x Breadth x Height