Who forms pyramids and raps circles around square lyricists?

Eminem, Cinderella Man.

Eminem’s latest (and greatest?) album, Recovery, has defied convention in many ways:

- Few rappers have enjoyed success well into their 30s or more than a decade into their careers, but Eminem has led the Billboard charts now five weeks and counting.
- In the digital age, few artists can even sell full albums anymore, but Recovery sold more albums in its first week than any album since 2008.
- In between aggressive, venomous lyrics that need to be rewound to be fully appreciated, Eminem also provides listeners with an insider knowledge of some of the GMAT’s unconventional geometry questions.

With his lyric “who forms pyramids and raps circles around square lyricists”, the world’s greatest inside-out rhyme pattern rapper isn’t merely calling out the world’s other emcees, but also calling attention to one of the GMAT’s favorite inside-out Geometry question formats, those that wrap circles around squares and vice versa.

The GMAT loves to ask questions about:

A circle inscribed in a square

and

A square inscribed in a circle

This allows the exam to test your ability to link together seemingly disparate concepts. But much like a top emcee, you can blend together unique concepts to achieve financial success.

When a square is inscribed inside a circle:

The diagonal of the square is the same length as the diameter of the circle, which also means that the diameter of the circle / diagonal of the square takes the “x * squareroot 2″ side of the 45-45-90 triangle that it creates with two sides of the square. Divide the diameter by the square root of 2 and you have the length of the side of the square.

In this case, just having one piece of information allows you to solve for everything else. Given the area or circumference of the circle, you can find the radius and then the diameter, and use the diameter to acquire the lengths of the sides of the square. Given the area or perimeter of the square and you can find the diagonal, use that as the diameter of the circle, and find out whatever you need to know about the circle that way.

When a circle is inscribed inside a square:

The diameter of the circle is the same as the length of a side of the square. This may make things even easier than the square-inside-circle setup. Once you have the area, circumference, or radius of the circle, you can find the diameter and therefore a side of the square. Or, if given information about the square, you can then find the length of a side, equate it to the diameter, and use that to unlock the numerical properties of the circle.

Circles and squares are perfect shapes because they’re perfectly symmetrical: find one piece of information about either and you know about the entire thing. Because of this, they’re great shapes for the GMAT to use to test your ability to link together concepts, and these one-inscribed-in-another questions have long been favorite geometry question fodder. Learn to wrap circles around squares and squares around circles and you’ll be testing circles around the GMAT in no time. The next step after that? Echoing Eminem’s “Hi, My Name Is…” at business school orientation.

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