Multiple Figures in GMAT Geometry

October 01, 2007

Some of the most difficult GMAT quantitative questions are geometry problems with multiple figures in the diagram. Odds are you know just what I'm talking about: a triangle inside a circle, a circle inscribed in a square, three rectangles arranged in a bigger rectangle...the possibilities are endless.

Questions like these can be overwhelming. Every additional figure not only gives you that shape to worry about, but also all of the interactions between the two shapes. In addition to the more traditional techiques you'll use when you see, for instance, a circle, there are two tips I can share for questions like these.

First, assume that everything in the diagram is relevant. Second, connect the dots.

Everything is Relevant

On GMAT Problem Solving questions, there is almost never superfluous information in the question or diagram. You can use this to your advantage.

For example, let's say the diagram is a semicircle with several lines drawn from the center of the semicircle to points on the circumference of the semicircle. More lines are added to create triangles out of those lines. Most people, seeing a diagram like that, focus on the triangles. It's natural: triangles are a key point in most GMAT geometry problems.

But there's a semicircle in the diagram, too! That's no accident. All of those lines drawn from the center of the semicircle to the circumference are radiuses. Thus, all of the triangles formed from those lines are isoceles. Helpful, huh?

Connect the Dots

Especially when figures are inscribed in each other, their relationships are not immediately clear. A reasonably simple example is a square inscribed in a circle. The diagonal of the square is the diameter of the circle, so if you know something about one of the figures (perhaps the area of the square), you can figure out the dimensions of the other one.

But in most cases, you'll have to find the connection. The GMAT isn't going to give you that diagonal.

Consider a more difficult example. What if a circle is inscribed in an equilateral triangle? Not so straightforward, but definitely the sort of thing that could be the difference between a 740 and a 750 GMAT score. I'm going to let you work on that example on your own, but I'll tell you this much: there is a connection. If I gave you the area of the circle, you have enough information to find, say, the perimeter of the triangle.

I'll limit myself to the single hint of this section: you'll have to connect the dots. Draw the diagram (on paper--in your head doesn't count) and see what kind of relationships you can discover between the two figures. I'll post an answer on this site tomorrow.

About the author: Jeff Sackmann is a GMAT tutor based in New York City. He has created many resources for GMAT preparation, including the popular Total GMAT Math and Total GMAT Verbal, as well as 1,800 practice GMAT math questions.

• Older: Techniques for GMAT Sentence Correction Practice
• Newer: Circle Inscribed In an Equilateral Triangle
• General GMAT Math: more articles
• All articles at GMAT HACKS