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GATE DA 2025 Q8: Dodecagon triangles rearranged into squares of side r

Question

A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius rr cm as shown in the figure. The side of the dodecagon is dd cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side rr 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.

A regular dodecagon inscribed in a dashed circle, divided into 12 numbered triangles; one side is labelled d.
Tap the figure to enlarge

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Answer: A (official GATE 2025 final answer key)

Concept used: Area of a regular dodecagon inscribed in a circle of radius r is 3r23r^2.

First step: The dodecagon's area is 12×12r2sin⁡30∘=3r212 \times \tfrac12 r^2 \sin 30^\circ = 3r^2, while each square has area r2r^2. Compare the two areas.

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key

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