Open the Box
Weekly Problem 37 - 2015
A piece of card is folded to make an open box. Given its surface area, can you work out its volume?
A piece of card is folded to make an open box. Given its surface area, can you work out its volume?
Age
11 to 14
Challenge level
Problem
Image
A square piece of card has a square of side $2$ cm cut out from each of its corners. The remaining card is then folded along the dotted lines shown to form an open box whose total internal surface area is $180$ cm$^2$.
What is the volume of the open box in cm$^3$?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
The original square has area 180 + 4 x 4 = 196, so it has side length 14cm.
Therefore, the dimensions of the box are 10 x 10 x 2.
Therefore, the volume is $200$cm$^3$.
Alternatively:
The base of the open box is a square. Let its side be of length $x$ cm.
Then the total surface area of the box in cm$^2$ is $x^2+4\times 2x=x^2+8x$.
Hence, $x^2+8x=180$, that is $x^2+8x-180=0$.
Therefore, $(x+18)(x-10)=0$ which gives $x=-18$ or $x=10$.
As $x$ must be positive, it must be the case that $x=10$.
Now the open box has dimensions $10$ cm $\times 10$ cm $\times 2$ cm. So its volume is $200$cm$^3$.