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This is a picture of the tiles on our bathroom floor a while ago. I think they look nice, but laying them caused quite a lot of trouble, largely for mathematical reasons.
How many tiles do you need to cover the floor of a more-or-less rectangular bathroom? If the tiles are square or rectangular, it's not too hard to work it out: the area of the tile is just the width times the height, so you divide the floor area by the area of one tile, and add on a few more for luck. But what's the area of a hexagon?
The easiest way to work it out is draw a straight line, shown in red here, from each of the vertices (sharp bits) of the hexagon to the centre. Now you have six equilateral triangles. Each of them takes up one of the sides (whose length we'll call "a") and they meet in a point at the centre.
Next, you draw a line from the midpoint of each of the sides of the hexagon to the centre, as shown in blue. Now you have twelve right-angled triangles. The shortest side of each of those triangles (in black, taking up half of one edge of the hexagon) has length ½a, and the angle beside it (at the vertex) is 60 degrees. So the side perpendicular to the shortest side, in blue, going from the midpoint of the edge to the centre of the hexagon, has length ½a tan 60°. And tan 60° = √3. So that side has length √3/2 a.
The black and blue edges are perpendicular, so the area of a triangle is the product of their lengths, divided by two because the triangle is half of a rectangle. That comes to ½a, times √3/2 a, divided by two, which is √3/8 a². And we have twelve right-angled triangles, so the area of the hexagon is 12√3/8 a², or 3√3/2 a².
So we just had to measure the length of a side of one of our hexagonal tiles, plug it into that formula, divide the bathroom floor area by the result, and that's the number of tiles we needed. Job done, right?
Unfortunately not. When our bathroom fitter arrived, he explained to us that although we theoretically had enough tiles, he needed more, because the floor was not a whole number of tiles wide, and the tiles could only be split in certain ways. So my wife had to make a forty mile round trip to buy one extra box of tiles. Hurrah for the open road.
All this shows that maths is easier than fitting bathrooms. It's also useful in a whole range of careers beyond bathroom fitting, so do contact me if you would like some A-level tutoring help.
By the way, if you ever find yourself having to decide on tiles for a bathroom floor, wouldn't it be cool to choose them in two contrasting colours, in just the right sizes to make up a Sierpinski carpet? I'm sure everyone in the household would love it, though your bathroom fitter might have another opinion.
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