Question

What is the value of the ratio\( \frac{\text{Area of the hexagon}}{\text{Area of the grey triangle}}\)?

Hint:

For geometry problems involving regular polygons, decomposing the shape into smaller, simpler shapes (like equilateral triangles in a hexagon) is a very effective strategy. Also, using a coordinate system can simplify area and length calculations.

Answer 1

Step 1: Place the hexagon and mark the three points.
Put the hexagon's centre at the origin with vertex A at \( (s, 0) \). Going around by \( 60^{\circ} \) steps gives B at \( \left(\frac{s}{2}, \frac{s\sqrt{3}}{2}\right) \), and the opposite vertex D at \( (-s, 0) \).

Step 2: Use the shoelace (determinant) formula on A, B, D.
\[ \text{Area}_{ABD} = \frac{1}{2}\left| x_A(y_B - y_D) + x_B(y_D - y_A) + x_D(y_A - y_B) \right| \]

Step 3: Substitute the coordinates.
\[ = \frac{1}{2}\left| s\left(\frac{s\sqrt{3}}{2} - 0\right) + \frac{s}{2}\left(0 - 0\right) + (-s)\left(0 - \frac{s\sqrt{3}}{2}\right) \right| \]
\[ = \frac{1}{2}\left| \frac{s^2\sqrt{3}}{2} + \frac{s^2\sqrt{3}}{2} \right| = \frac{s^2\sqrt{3}}{2} \]

Step 4: Divide by the hexagon's area.
\[ \frac{\left(\frac{3\sqrt{3}}{2}\right)s^2}{\left(\frac{\sqrt{3}}{2}\right)s^2} = 3 \]
\[ \boxed{3} \]

Answer 2

A third approach uses the long diagonal AD to split the hexagon into two congruent trapezoids, without any trigonometry or coordinates.

The diagonal AD splits the regular hexagon into two congruent trapezoids, so each trapezoid, including ABCD, has exactly half the hexagon's area: \( \frac{3\sqrt{3}}{4}s^2 \).

Triangle ABD sits inside trapezoid ABCD, and equals that trapezoid's area minus the small corner triangle BCD. Triangle BCD has two sides of length \( s \) meeting at the hexagon's own interior angle of \( 120^{\circ} \) at vertex C, so its area is \( \frac{1}{2}(s)(s)\sin(120^{\circ}) = \frac{\sqrt{3}}{4}s^2 \).

So the area of ABD is \( \frac{3\sqrt{3}}{4}s^2 - \frac{\sqrt{3}}{4}s^2 = \frac{\sqrt{3}}{2}s^2 \).

Comparing this with the hexagon's area \( \frac{3\sqrt{3}}{2}s^2 \) gives a ratio of 3. So the correct answer is 3.