S98-20

Bunuel,

I am confused at the step in which you derive the height side of the first 30-60-90 triangle. the "x" side is r/2, because the "2x" side is r, but why is "x sqrt 3" side not = to (r sqrt 3)/2? instead you have it divided by 4? I must be missing something. You also say that side AX = XB? That can't be because we just found it to be a 30-60-90 right triangle. You must've meant to say that AX = AX, thus we know AC?

The following discussion might help: approximately-what-percent-of-the-area-of-the-circle-shown-i-169744.html

Official Solution:


Approximately what percent of the area of the circle shown is shaded, if polygon ABCDEF is a regular hexagon?




A. 24%
B. 30%
C. 36%
D. 42%
E. 48%


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 \(180(n - 2)^\circ = 180(6 - 2)^\circ = 720^\circ\), and there are 6 interior, equal angles, then each of those angles must measure \(\frac{720^\circ}{6} = 120^\circ\).

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 \(60^\circ\), so \(AXB\) is a 30-60-90 triangle. This means that \(XB\) is \(\frac{r}{2}\) in length, and \(AX\) is \(\frac{\sqrt{3}r}{4}\) in length. Since \(AX\) and \(XB\) are the same length (by symmetry), \(AC\) is \(\sqrt{3}r\) in length.

This means that the area of triangle \(ABC\) is \(\frac{1}{2} \leftarrow( \sqrt{3}r \rightarrow) \leftarrow( \frac{r}{4} \rightarrow) = \frac{\sqrt{3} r^2}{4}\). Since there are three shaded triangles in all (including \(ABC\)), the total shaded area is \(\frac{3 \sqrt{3} r^2}{4}\). The area of the circle is , so the ratio of the shaded area to the area of the circle is given by
\(\frac{3 \sqrt{3} r^2}{4} \div \pi r^2 = \frac{3 \sqrt{3}}{4 \pi} \approx \frac{\sqrt{3}}{4} \approx \frac{1.7}{4} \approx 0.43\)

The closest answer is 42%.


Answer: D