Chapter 12

Circles & Cylinders

The Basic Elements of a Circle

A circle is a set of points that are all the same distance from a central point. By definition, every circle has a center. Although the center is not itself a point on the circle, it is nevertheless an important component of the circle. The radius of a circle is defined as the distance between the center of the circle and a point on the circle. The first thing to know about radii is that any line segment connecting the center of the circle (usually labeled O) and any point on the circle is a radius (usually labeled r). All radii in the same circle have the same length:

You will review the other basic elements by dealing with a particular circle. Your circle will have a radius of 7, like the one shown here, and you’ll see what else you can figure out about the circle based on that one measurement. As you’ll see, you’ll be able to figure out quite a lot.

Once you know the radius, the next easiest piece to figure out is the diameter. The diameter passes through the center of a circle and connects two opposite points on the circle:

One way of thinking about the diameter (usually referred to as d) is that it is 2 radii laid end to end. The diameter will always be exactly twice the length of the radius. This relationship can be expressed as d = 2r. That means that your circle with radius 7 has a diameter of 14.

Now it’s time for the next important measurement—the circumference. Circumference (usually referred to as C) is a measure of the distance around a circle. One way to think about circumference is that it’s the perimeter of a circle:

As it happens, there is a consistent relationship between the circumference and the diameter of any circle. If you were to divide the circumference by the diameter, you would always get the same number—3.14 … (the number is actually a non-terminating decimal, so it’s usually rounded to the hundredths place). You may be more familiar with this number as the Greek letter π (pi). To recap:

OR πd = C

In your circle with a diameter of 14, the circumference is π(14) = 14π. The vast majority of questions that involve circles and π will use the Greek letter rather than the decimal approximation for π. Suppose a question about your circle with radius 7 asked for the circumference. The correct answer would read 14π, rather than 43.96 (which is 14 × 3.14). It’s worth mentioning that another very common way of expressing the circumference is that twice the radius times π also equals C, because the diameter is twice the radius. This relationship is commonly expressed as C = 2πr. As you prepare for the GRE, you should be comfortable using either equation.

There is one more element of a circle that you’ll need to be familiar with, and that is area. The area (usually referred to as A) is the space inside the circle:

Once again, it turns out that there is a consistent relationship between the area of a circle and its diameter (and radius). The formula for the area of a circle is A = πr2. For your circle of radius 7, the area is π(7)2 = 49π. To recap, once you know the radius, you are able to determine the diameter, the circumference, and the area:

C = 14π
A = 49π

These relationships are true of any circle. What’s more, if you know any of these values, you can determine the rest. In fact, the ability to use one element of a circle to determine another element is one of the most important skills for answering questions about circles. To review:

To demonstrate, you’ll work through another circle, but this time you know that the area of the circle is 36π. Using the formula for the area, start by plugging this value into it:

36π = πr2

Now, solve for the radius by isolating r:

36π = πr2
  36 = r2
    6 = r		 Divide both sides by π.
Take the square root of both sides.

Now that you know the radius, you can multiply it by 2 to get the diameter, so your diameter is 6 × 2, which is 12. Finally, to find the circumference, multiply the diameter by π, which gives you a circumference of 12π:

Check Your Skills

  1. The radius of a circle is 7. What is the area?

  2. The circumference of a circle is 17π. What is the diameter?

  3. The area of a circle is 25π. What is the circumference?