Discussion
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius r cm as shown in the figure. The side of the dodecagon is d cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side r cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
Answer: A
The answer is 3 squares, each made from 4 triangles.
A regular dodecagon inscribed in a circle of radius r is made of 12 central triangles, each with two sides r and apex angle 30°:
\[ \begin{aligned} \text{Area} &= 12 \times \tfrac{1}{2}r^{2}\sin 30^{\circ} \\ &= 3r^{2} \end{aligned} \]
So the 12 triangles together have the area of exactly 3 squares of side r, and using every triangle once gives 3 squares of 12/3 = 4 triangles each. In the figure each square is made of one equilateral triangle of side r (such as 1), two 30° isosceles triangles (such as 2 and 3) and one of the thin edge triangles (10, 11, 12):
\[ \begin{aligned} &\tfrac{\sqrt{3}}{4}r^{2} + 2 \cdot \tfrac{1}{4}r^{2} \\ &+ \left(\tfrac{1}{2} - \tfrac{\sqrt{3}}{4}\right)r^{2} = r^{2} \end{aligned} \]
Four squares would need area \(4r^{2}\), and 3 or 2 triangles per square would leave triangles unused.
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