Welcome to “Land Your Score,” a blog series in which Kaplan instructor Jennifer Land shares key insights and strategies for improving your GMAT performance on Test Day. This week, Jennifer discusses using circle ratios to solve geometry problems in the Quantitative Reasoning section.
GMAT geometry problems often involve circles. The standard circle concepts (radius, diameter) and equations (area, circumference) are essential to know on Test Day, but there is one additional layer of knowledge that is essential to answering higher-difficulty circle problems.
Circle components in GMAT geometry problems
Circles found in the Quantitative Reasoning section are often drawn with a central angle marked; in the figure below, the central angle is X. Angle X measures n°. Points A and B along the circle’s circumference represent the arc. Imagine that this circle is a pizza and the shaded area represents one slice. The arc is the crust on one slice of the pizza. The slice itself, bordered by the arc and the central angle, is called a sector.
The measure of the central angle represents a fraction of the measure of the whole circle (360°), and the length of the associated arc represents the same fraction of the circumference (2πr). The area of the sector is also the same fraction of the area of the circle (πr²).
Looking at the circle above, let’s say n=60°. What fraction of the entire circle is that?
This central angle is ⅙ of the circle. Using the pizza analogy, if a pizza is cut into 60° slices, it is cut into 6 slices. The area of one slice is ⅙ of the area of the entire pizza. Its arc represents ⅙ of the circumference of the pizza, or ⅙ of the crust.
Circle component ratios
These three fractions can be written as follows:
Together, they form the circle ratio that you need to know for some of the GMAT’s geometry problems.
Now let’s go back to our circle above and take it further. Let’s say the diameter of the circle is 12. That means the radius is 6, the circumference is 12π, and the area of the circle is 36π. We can use these values to solve for the arc length by plugging the numbers into the ratio:
Solving for arc length by cross-multiplying and dividing, we find arc = 2π. We can then do the same thing to find the area of the sector:
Solving for the sector area by cross-multiplying and dividing, we find sector = 6π.
Quantitative reasoning through analogy
The pizza analogy really helps me (and my students) remember these ratios. The degree measure of the tip of one slice is a fraction of 360. The slice of pizza is the same fraction of the entire pizza, and its crust is the same fraction of the entire crust. In this example, each component was ⅙ of the total.
Next week, I’ll share some advice for spotting common trap answers in Data Sufficiency questions. In the meantime, you can wow your friends at the next pizza party—or at least use your next pizza break as an opportunity for GMAT prep.
Want to master geometry problems? Explore our GMAT prep course options and class schedules.
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