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The Math Hidden In Your Living Room

Mar 28, 2008
A question that vexes math students and teachers alike - "How does this apply to the rest of my life?" - turns out to have some surprising answers. Geometry in the living room? Statistics in your ledger? Yes, and yes.

One place where math affects almost everyone, of course, is the pocketbook. Anyone who has undertaken a home decorating or remodeling project knows just how much our plans are constrained (and sometimes inspired) by the need to stay within budget! What many people don't know is that a little knowledge of geometry can help you do exactly that.

Laying carpet is one of the basic home-decorating tasks, and it becomes immensely easier when you keep in mind the formula we all learned in sophomore geometry: A = L x W (area equals length times width). By measuring the length and width of your floor, then multiplying them against each other, you find out exactly how much carpet you need.

But wait! What if you live in a geodesic dome? (Well, it could happen.) Or, more likely, what if you have a circular alcove at the end of one room? Most of us learned the formula for figuring a circle in high school (and many of us then, having taken the test, promptly forgot it), but here it comes to the rescue: A = (pi) X r2 (area equals pi times the radius squared). "Radius" is half the circle's diameter (its length at its widest part), and "squared" simply means multiplying a number by itself.

Pi, meanwhile, is a number with its own fascinating history, but all you need to know is that it's more or less equal to 3.14159. (If I were to write out all the numbers that follow the decimal point, we'd be here for awhile: mathematicians using computers have plotted out pi to millions of digits, and they're not done yet.)

Let's say you have a home office with a rounded alcove at the end. To figure out how much carpet you'll need here, first of all, figure out the carpet needed for the rest of the room as if the alcove weren't there: measure length and width up to the point where the semicircle begins, and figure out the area of the rest of the room as described above.

Let's say your room is 18 feet wide by 10 feet long, leaving out the alcove - you need, then, 18 times 10 feet worth of carpet, or 180 square feet. Keep this figure in mind.

Now, to measure the semicircular part of the room, you need, first of all, to know the diameter of the circle involved (so you can figure out the radius). "Diameter" is the width of the circle at its widest point, which in this case is the base from which the circle bulges outward - the length of your living room, in other words. Measure the length of your living room, then halve that to get the radius (the r2 in the formula mentioned above).

If your living room is 18 feet wide, r will then be nine feet. "Square" that which means, multiply it by itself: 9 X 9 gives you 81. The area of the circle, then, is pi - 3.14159 - times 81. Plugging in the numbers, we get about 254 feet. (Actually, we get 254.46879 - but for our purposes it's perfectly OK to round up or down to get ride of the numbers after the decimal point.)

Now, that number "254" would be the amount of carpet you needed if you had a whole circle to carpet. (So if you are living in that geodesic dome, you can go order 254 square feet of carpet now.) But for our example, we had only a semicircle - a little bulging alcove at the end of an otherwise-square room. So you'll only need half of this, because you only have a half circle. Half of 254 is 127 square feet - so that's how much carpet you'll need just for this alcove at the edge of the room. Add this much to the amount of carpet you need 180 square feet, in this example. 180 plus 127 is 307 square feet.
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