The figure shows a circle of radius r whose diameter has been cut in half, giving two smaller circles of radius r/2 and r/4 as shown. Calculate the approximate circumference of the outer circle and the approximate area of the inner (white) region. Express your answer in terms of r, to at least two decimal places where applicable, and with an appropriate degree of precision. Your answer will be rounded to the nearest whole number if needed; it should not be rounded up or down to an even number in this case.
The problem at hand
Comparing two circles when there is not a perfect way to measure their circumferences without using a protractor or similar tool. If you are asked to find such an estimate, your first impulse might be to say that you can’t—that there’s no possible way to compare measurements that don’t have an agreed-upon scale. But in fact, with only a ruler and a keen eye, it can be done.
A bit about circles
A circle’s circumference—the distance around it—is usually calculated using a simple formula: multiply π (3.14159…) by twice the diameter, then add that number to itself once. In other words, you multiply 3.14 by 2 times your diameter and add that to 3.14 plus 1, which equals π (3.14159…). What about inside a triangle?
How do we solve this problem?
We solve it using some geometry. We start by finding a diameter which we call D. In order to find D, we use a formula from geometry called the Pythagorean theorem.
Formulating a solution
The formula for calculating circumference is C = 2πr. Substituting π with 3.14, you get (2)(3.14)r = 2(9.52)(r). This means that each leg of your triangle must be 9.52 units long to have a 9-inch circumference. Since there are two legs, your hypotenuse must be twice as long, or 19.04 inches long!
Why did this method work?
We know that for every two full revolutions around a circle, we have traveled a total distance equal to πr. The first approach wasn’t exactly accurate since it involved dividing by 6.28, which isn’t an integer. However, it gave us a fairly good estimate (an overestimate). Since we wanted to get an exact answer, however, we divided by 100 so that our estimation could be perfect. Why did that work?
This applies to more than circles
The formula for finding out how much yarn you will need to crochet a project can be useful with other craft projects as well. For example, if you wanted to make a scarf, you could use your favorite free scarf pattern and find out how much yarn it requires.
How do you find the approximate circumference of a circle?
Measure each diameter, then divide by 2. Then, you multiply that number by pi (3.14) and get your answer. For example, if a line measures 10 inches long, then there are two diameters. Each diameter is 5 inches long. So 5/2 = 2 1/2 and then 2.5 x 3.14 = 7-ish? I think?
What is the meaning of the circumference of a circle?
an expression (such as an equation) that indicates what a number or variable represents. The circumference of a circle is simply a measure of how big that circle is, measured in distance. It can be expressed as an exact value (usually given in meters), but there are also approximations that can provide us with more useful information – one example would be calculating your waist size.
It takes 2,784 inches to make a 360° circle. If you know that there are 12 inches in one foot and 6 feet in one yard, then you will know that it would take approximately 656 yards to finish off a 360° circle. That’s 3 football fields worth of dough! Alternatively, if you put enough doughnuts on top of each other, it might be possible to complete a circle around your waist!