An Introduction to Pi and Its Use in Circle ProblemsDate: 04/28/2004 at 22:46:36 From: josh Subject: i have lots of trouble with pi problems will you please help I don't understand problems that involve pi. For example, if the circumference of a circle is 66 mm, what is the radius? I just don't understand what pi is and how you use it. Date: 04/29/2004 at 09:38:19 From: Doctor Ian Subject: Re: i have lots of trouble with pi problems will you please help Hi Josh, One of the most common places that we run into pi is in dealing with circles. For any circle, there are a number of things we can measure or calculate. These include: 1. The circumference This is the distance around the circle. That is, if we make a mark on the circle, then go around the circle until we get back to the mark, the distance we travel is the circumference. 2. The radius Every point on a circle is the same distance from the center of the circle. This distance is called the radius of the circle. That is, if we start at the center and move to any point on the circle, the distance we travel is the radius. 3. The diameter The diameter is just twice the radius. Another way to think about it is that if we pick any point on a circle, the distance to the point that is farthest away--directly on the opposite side of the center--is the diameter. 4. The area Area is different than distance, in that it's two-dimensional, where distance is one-dimensional. If you're not clear on the difference between the two, take a look at Area and Perimeter http://mathforum.org/library/drmath/view/57652.html Okay, so what does pi have to do with all this? Well, if we pick _any_ circle, and we divide the circumference (the distance around) by the diameter (the distance across), the result is _always_ pi: circumference distance around ------------- = --------------- = pi diameter distance across Note that for this to work out, we have to measure both distances in the same units! (For example, if we have the circumference in feet, and the diameter in inches, the ratio will not be equal to pi.) The important thing to understand is that ANY circle, no matter how big or small, always gives the same answer when you divide its circumference by its diameter. As the circle gets bigger or smaller, the circumference and diameter both increase or decrease together, and their ratio stays the same. This constant ratio or division answer is what we call "pi". There are several approximations for the actual value of pi that are commonly used. One of them is 22/7, others are 3.14 or 3.14159. Remember that all of these are just approximations, as the actual value of pi has an infinite decimal expansion that does not go into a repeating pattern. Pi starts out 3.14159... so that's a good approximation but still not completely accurate. 3.14 is less accurate, and 22/7, which when divided gives 3.142857... is also less accurate. Since pi always has the same value, if we know either the circumference or the diameter of a circle we can find the other, by writing the above equation with the values we know, and solving for the one we don't. For example, suppose we know that the diameter of a circle is 12 cm, and we want to know the circumference: circumference pi = ------------- 12 cm If we multiply both sides of this equation by 12 cm, the denominator on the right cancels out: circumference pi * 12 cm = ------------- * 12 cm 12 cm circumference * 12 cm pi * 12 cm = --------------------- 12 cm 12 cm pi * 12 cm = circumference * ----- 12 cm pi * 12 cm = circumference * 1 pi * 12 cm = circumference Now, we can just leave it like this, or we can use an approximate value for pi to get an approximate value for the circumference, e.g., 3.14 * 12 cm = 37.68 cm Okay, so what if we know the circumference instead, and we want to get the diameter? For example, suppose we know that the circumference is 42 cm. Then we have 42 cm pi = -------- diameter Doing the same sort of thing as before, we can multiply both sides by the diameter, to get pi * diameter = 42 cm and then divide both sides by pi to get 42 cm diameter = ----- = (42/pi) cm pi Again, we could go ahead and use an approximate value for pi, diameter = (42 / 3.14) cm = 13.38 cm Does this make sense so far? The relationship between diameter and radius is much simpler. Recall that the diameter is just twice the radius, which means that the radius is half the diameter. So to get one from the other, we just multiply or divide by 2. Why do we care about the radius? Well, to find the area of a circle, we need to use a second formula: area = pi * radius^2 = pi * radius * radius So a typical problem might be: The circumference of a circle is 42 cm. What is the area of the circle? To solve this, we would use the circumference to get the diameter, as we did above: 13.38 cm. Then we would divide it by two to get the radius: 6.69 cm. Then we would plug that into the formula for area: area = pi * 6.69^2 = 3.14 * 140.5 cm^2 Were you able to follow that? (Let me know if any of this has been unclear.) This might be a good time to point out that although it's tempting to just plug in an approximate value for pi, especially when you have a calculator handy, there are good reasons for avoiding that for as long as possible. Let's look again at the problem we just solved. We're told that the circumference is 42 cm. That means the diameter is 42/pi cm. And the radius is half that, or 21/pi cm. So the area is area = pi * (42/pi)^2 42 42 = pi * -- * -- pi pi pi * 42 * 42 = ------------ pi * pi 42 * 42 = ------- Two of the pi's cancel! pi So now we just enter pi once at the end, and we don't have to worry about errors introduced by using an approximate value in the middle. Since pi shows up in lots of different formulas, there's a pretty good chance that in any problem with multiple steps, a pi that shows up in one step will get canceled by a pi that shows up in a later step. So by just leaving it in there, you can save yourself work, and reduce the opportunity to introduce errors. Just to show how you can go in the other direction, let's try starting with the area instead of the circumference: The area of a circle is 100 cm^2. What is the circumference of the circle? Now we start with the same formula, area = pi * radius^2 100 = pi * radius^2 We divide both sides of the equation by pi to get 100/pi = radius^2 What now? We take the square root of both sides: ______ ________ \/100/pi = \/radius^2 ___ \/100 ----- = radius __ \/pi 10 ----- = radius __ \/pi I hope this helps! Write back if you have more questions, about this or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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