There are only three formulas you will need to solve circle problems.
The radius is involved in all three formulas. Once you have the radius of a circle, you will know almost everything there is to know about that circle.
πr2 measures the area of a circle. It’s easy to remember because area, such as the area of a house or apartment, is always measured in units squared.
πd or 2πr measures circumference. If you know circumference, you know the radius, and if you know the radius you know the area. Most GRE circle questions ask you to find one or the other or require you to convert from one to the other. You must be able to do these tasks quickly and easily. If you write the formulas down on your scratch paper and fill in the information from the question directly underneath the relevant part of the formula, finding the answer shouldn’t be a problem.
is one formula that they don’t give you in any of the official GRE literature, but it can comes in handy. It essentially means that angles, arcs, and areas are all proportional. If you were to divide a circle into quarters, the central angle—90 over 360—reduces to 
. The
resulting arc is of the circumference of the circle and the area of the sector is the area of the circle.
Pi, or π, equals 3.14159 … or 3 and change. If you are given a circle with a radius of 5 and asked for the area, set π equal to 3 and Ballpark. Eliminate any answer choice which is less than or equal to 75, or greater than or equal to 100. You know that the correct answer will be far closer to 75 than it will be to 100.
The five-step approach to geometry problems applies to circles as well.
In some cases the test will give you a shape, which you may or may not be able to trust, and in others it will give you a word problem and leave it up to you to envision the shape. As with every other part of the test, getting your hand moving is an important first step to entering the problem. Get your shape down on your scratch paper so that you can begin working with it there. On Quant Comp questions involving geometry, instead of Plugging In more than once, you may have to draw your shape more than once.
Whether you are given the shape or not, you will be given a certain amount of information regarding the shape such as the measure of some angles, lengths of some sides, areas of some sides, or volume. Put that information in the figure.
If you are given two angles of a triangle, find the third. If you are given the radius of a circle, find the area. Often this will be the entire problem. Geometry on the GRE is all about finding the missing piece of information. You will be given just enough information to find the piece that is missing.
If step three didn’t get you the answer, you must still be missing a piece of information. Writing down the formula is a way of both organizing your information and telling you what is missing. When you write your formulas down, fill in the information you have directly underneath the relevant part of the formula. It seems simple, but this way you can’t make a mistake. Finding the missing piece of information becomes a simple case of solving for x.
If you’re still stuck, you may need to manipulate or subdivide your circle into smaller shapes. If create triangles, draw in the height. Have you created a 30-60-90? A 45-45-90? Or a Pythagorean triple? Try subdividing the shape or, if it’s a three-dimensional figure, dashing in the hidden lines.
Often, you will see circles in combination with other shapes. If you don’t immediately see the correct path to the solution, look for the radius. Everything about a circle derives from there. It is possible that you will see a circle inscribed on a coordinate plane. The same rules apply. Use right triangles to find the end points of as many radii as you need to check the answer choices that you can’t eliminate through Ballparking.
For more practice and a more in-depth look at The Princeton Review math techniques, check out our student-friendly guidebook, Cracking the GRE.