Circles Tutorial for SAT/GRE Test Preparation

Circles are beautiful figures that have for a long time inspired mathematicians and philosophers. The defining feature of a circle is perfect symmetry about a center point.

The only thing you need to know about a circle is how far it is to the edge from the center. This is called the radius . Because of the symmetry, it is the same straight-line distance from the center to any point on the edge of the circle. And once you know the radius, you'll also know both the circumference and the area.

Sometimes you'll be given the diameter rather than the radius. The diameter is a straight line distance across the circle, passing through the center. So the diameter just is twice the radius. Once you know one, you know the other.

There are formulas for the circumference and area of a circle. They involve a very special number called pi - designated by . = 3.14159... . The "..." means that the number keeps going forever. It really does! People have written it out to millions of digits, and there are millions of millions more to go. It is a number that never ends.

Now may seem like a weird thing to all of a sudden show up in these formulas. We need it because while it is easy to imagine taking a ruler to measure the straight edges of a rectangle or square, it is not so easy to think about using a straight edge to measure a rounded contour. A long time ago, some very clever people figured out that you could pretend that the circles edge was actually a bunch of very small straight edges connected together, and then you can think about using a ruler. And to express the process of thinking about smaller and smaller edges, they needed to invent .

Just trust us that is a very important and useful (and mysterious) number.

The circumference of a circle is twice the length of the radius times

Perimeter = 2 × × r

The area of a circle is times the length of the radius squared.

Area = × r²

(Notice that area is always a squared unit of measure. If you ever confuse the two formulas, stop and think: "Okay, area needs a square, so area must be × r².)

Because r is the only thing you need to compute about a circle to know everything about it, once you have the area or circumference of a circle you know the radius. Try the following:

1. What's the radius of a circle with area 9 ?
2. What's the radius of a circle with perimeter 12 ?
3. What is the area of a circle with perimeter of 42 ?

Answers
1. Use the formula relating the area of a circle to its radius. The radius squared is 9, so the radius must be 3.
2. Look at the formula relating the perimeter to the radius. The radius of this circle is 12 divided by 2, or 6.
3. You can figure out the radius just like the previous question: the radius here is 21. Now plug into the formula for the area.... (You can do it!).

Cylinders
There are a couple of ways circles are used to form three-dimensional figures. One is the cylinder, which is just a circle with an added dimension of height.

The volume of a cylinder is the area of the circle times the height.

Volume = [Area of circle] × Height
= × r² × h

Spheres
Spheres appear rarely on these kinds of standardized tests. If the symmetry of the two dimensional circle is extended to three dimensions, you get a sphere. (A soccer ball is a sphere.)

The volume of a sphere is 4/3 × r³.

Tangents
Before we leave circles, there's one other useful fact about them.

A line that just touches the circle at one point is called a tangent. The tangent line and the radial arm—the line from the center of the circle to point P—that touches point P form a right angle.

Tangents are cool. Imagine you hold a ball and spin around and around in a circle. If you suddenly let go of the ball, it will fly off in a direction tangent to the circle at the point you let it go. And this is exactly what going off on a tangent means, both when used literally and used to describe someone who strays from the main point of a conversation.

• Rectangles Tutorial
• Triangles Tutorial
• Lines Tutorial