While sat on the toilet I look at the pattern on the floor. It is a sequence of squares: xxx x1x xxx I wondered, how many x's surround square one, and what is the relationship between the squares in the middle and the surrounding squares? For example what if I chose the length of the squares in the middle to be two?: xxxx x12x x12x xxxx How many x's would surround the four squares in the middle? I have come to the conclusion the equation is four times the length of the middle squares plus the four corners: 4l+4 A square with a middle length of three has sixteen surrounding squares: (4×3)+4. xxxxx x123x x123x x123x xxxxx Does anyone else agree that this is correct?
Yes. Now, can you generalize it? Modify the formula so it works for rectangles. Modify the formula so that border width is independent of interior dimensions.
Your formula includes 4 x L. But that's because a square is 4 L's of the same length. So, for a rectangle, you just break that out into L and W - only two of each. So: Perimeter = (2 x L) + (2 x W) + 4 You can simplify this even more by gathering all the 2's: P = 2(L+W+2)
For a three-dimensional cube the formula would be 6 (4 l)+8 or 24l+8. If there were tiles covering the centre squares on a three-dimensional cube the formula would be 24l+6 (l^(2))+8?
