We’re launching a new weekly article: Challenge Problem of the Week! The question and solution to last week’s Challenge Problem will be posted here at Beat the GMAT and the current week’s question will be posted on the Manhattan GMAT web site. The problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 750+ level question, so do not worry if you cannot solve the problem in a 2 minute time frame. If you are up for the challenge, however, set your timer for 2 minutes and go!
Approximately what percent of the area of the circle shown is shaded, if polygon ABCDEF is a regular hexagon?
If polygon ABCDEF is a regular hexagon, then the figure is very symmetrical and actually looks like the drawing. We should avoid rigorous proofs and instead make quick arguments “from symmetry” – that is, recognizing that many parts of the diagram are equivalent.
We know that for the hexagon, each side is the same length and each interior angle is the same. Since the interior angles of a hexagon sum up to 180(n – 2)° = 180(6 – 2)° = 720°, and there are 6 interior, equal angles, then each of those angles must measure 720°/6 = 120°.
Moreover, each side of the hexagon is equal in length to the radius, since any regular hexagon can be chopped up into 6 smaller equilateral triangles, as shown by the lines in blue through the circle’s center O:
Consider small triangle AXB. The hypotenuse of this right triangle, AB, has length r. Angle ABO is 60°, so AXB is a 30-60-90 triangle. This means that XB is r/2 in length, and AX is in length. Since AX and XC are the same length (by symmetry), AC is in length.
This means that the area of triangle ABC is . Since there are three shaded triangles in all (including ABC), the total shaded area is . The area of the circle is , so the ratio of the shaded area to the area of the circle is given by .
The easiest way to simplify 1.7/4 is to estimate. 0.4 is 10% of 4. Therefore, 1.6 is 40% of 4. Because we started with 1.7, not 1.6, the answer should be just over 40%. The closest answer is 42%.
The correct answer is D.
To view the current Challenge Problem, simply visit the Challenge Problem page on Manhattan GMAT’s website.