Area of a Circle with Radius less than 1Date: 02/18/2002 at 19:11:11 From: Kelly Subject: Finding the Area od a circle with a radius less than 1 If the radius of a circle is less than 1, for example half an inch, it just gets smaller and you get a smaller area. I was doing a report on finding household objects and their circumference, area, etc., so I used inches on one small object, making it have radius less than 1. Then I tried with centimeters and I found that when I compared centimeters to inches by seeing how many centimeters fit into an inch, I found the length with the centimeters was longer because it was not less than 1. Please help! Thank you. -Kelly Date: 02/18/2002 at 19:29:15 From: Doctor Ian Subject: Re: Finding the Area od a circle with a radius less than 1 Hi Kelly, It doesn't really make sense to talk about a 'radius less than 1'. You need to specify some kind of unit of length: centimeters, inches, miles, meters, light years, and so on. If you pick the right units, you can make the radius of just about anything either less than or greater than 1. Consider a car tire. When measured in inches, the radius is clearly greater than 1. But when measured in miles, the radius is clearly less than 1. How can this be possible? The confusion clears up when you attach the units. It isn't surprising at all to say that something is 'larger than 1 inch but smaller than 1 mile', or 'larger than 1 foot but smaller than 1 yard'. Is it? If the point is to find areas of objects with radius less than 1, with no particular unit specified, then you can just choose a different unit for each object. When measuring a dinner plate, use yards. When measuring a quarter, use feet. When measuring a shirt button, use inches. And so on. Or just pick a really big unit, like yards, and measure everything in yards. You'll end up with some pretty silly numbers, i.e., a coat button is 1/2 inch in radius. So its radius in yards is 1 ft 1 yard r = (1/2) in * ----- * ------ = 1/72 yards 12 in 3 ft so the area is a = pi * r^2 = (22/7) * (1/72)^2 = 22/36288 square yards Does this help? Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 02/19/2002 at 19:20:04 From: Kelly Subject: Finding the Area of a circle with a radius less than 1 I don't know if that really answered my question. What I was trying to get at is that if a number is less than 1, it will be smaller than it is supposed to be. I think that 2.5 centimeters approximately equals 1 inch, so here is my example of what I mean: if you use a unit that is the same measurement but smaller, if you use something over 1 it will be larger even though it was the same at first - if that doesn't make sense look at my example. There is a circle. The Diamater is 1 inch, or 2.5 centimeters, and you want to find the area, so the radius is half an inch (.5 inch) or 1.25 cm. This is the work i did for each unit: Inches: Radius = .5 inch (.5 times .5 = .25) (.25 times 3.14 = .785) .785 inches transferred into centimeters should be around 1.9 (1.9625 exactly I think) but it is not, so if you do the centimeter work Centimeters: Radius = 1.25 cm (1.25 times 1.25= 1.5625) (1.5625 times 3.14 = approximately 4.9) 4.9cm is much more than the original 1.9 you would find. So I guess what I was wondering is why is this true? If a number is under 1 should you not square it? Is there some kind of formula explaining this? Is there something you can do to make me understand if there is some reason for this and if I am doing something wrong? Please reply! Thank you. -Kelly Date: 02/20/2002 at 12:33:05 From: Doctor Ian Subject: Re: Finding the Area of a circle with a radius less than 1 Hi Kelly, I think perhaps the problem is that you're confusing linear units (inches, centimeters, feet) with squared units (square inches, square centimeters, square feet). To use your example, if I have a circle whose radius is 1/2 inch, the area is a = pi * (1/2)^2 = (22/7)(1/2 in)(1/2 in) = 11/14 in^2 which is about 0.785 in^2. The exact conversion between inches and cm is 1 in = 2.54 cm, so when we use cm as the units, we find an area of a = pi * (2.54/2)^2 = (22/7)(2.54/2 cm)(2.54/2 cm) = (22 * 2.54 cm * 2.54 cm) / (7 * 2 * 2) = 5.07 cm^2 1.9 in^2 (which is another way to write '1.9 square inches'). So far, we're in agreement. And these two areas should be the same, right? That is, since it's the same circle, it ought to be true that 0.785 in^2 = 5.07 cm^2 and if I understand you correctly, you want to know (1) whether this is true, and if so, (2) why it's true. Well, let's look at the area of a square that is one inch on each side: 1 in +------+ | | 1 in area = 1 in * 1 in | | +------+ = (1 * 1) in^2 = 1 in^2 If we measure the sides in centimeters instead of inches, we have 2.54 cm +------+ | | 2.54 cm area = 2.54 cm * 2.54 cm | | +------+ = (2.54 * 2.54) cm^2 = 6.45 cm^2 That is, while it's true that 1 in = 2.54 cm it's NOT true that 1 in^2 = 2.54 cm^2 (Wrong!) Rather, the conversions are 1 in = 2.54 cm 1 in^2 = 6.45 cm^2 This might make more sense if we forget about the metric system for a moment. Let's look at square that is 1 yard on a side: 1 yd +----|----|----+ | | - - | | 1 yd - - | | +----|----|----+ The area is clearly 1 yd^2. But if we divide it up into square feet, 1 ft 1 ft 1 ft +----+----+----+ | | | | 1 ft +----+----+----+ | | | | 1 ft +----+----+----+ | | | | 1 ft +----+----+----+ we just as clearly have an area of 9 ft^2. Does this make sense? (It's easier to see what's going on in this case because it deals with whole numbers instead of fractions or decimals.) So the conversions are NOT 1 yd = 3 ft 1 yd^2 = 3 ft^2 (Wrong!) Rather, the conversions are 1 yd = 3 ft 1 yd^2 = 9 ft^2 Now let's get back to your original question. You want to know whether it's true that 0.785 in^2 = 5.07 cm^2 Well, let's use the conversion that we found, 1 in^2 = 6.45 cm^2: 6.45 cm^2 0.785 in^2 * --------- = 5.07 cm^2 1 in^2 5.06 cm^2 = 5.07 cm^2 which isn't exact, but the difference can be accounted for by the fact that we rounded off our decimal approximations. This is a subtle concept, and a lot of people have trouble with it, so don't feel too bad about being confused! You might want to take a look at Area and Perimeter http://mathforum.org/dr.math/problems/jessica.5.1.01.html to get a feel for how length and area differ from each other. I hope this helps. Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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