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GATE 2025 CY – Question 8

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

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.

Regular dodecagon divided into twelve center-to-edge triangles.
  1. 3; 4
  2. 4; 3
  3. 3; 3
  4. 3; 2

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Correct answer: (A) 3; 4

Explanation

The regular dodecagon is divided into 12 identical triangles by joining its centre to the vertices (each triangle has two sides equal to $r$ and the angle between them equal to $360^\circ/12=30^\circ$).

**Area of one triangle:**
$$A_\triangle=\tfrac12 r\cdot r\sin30^\circ=\frac{r^2}{4}.$$

**Triangles per square.** A square of side $r$ has area $r^2$, so
$$\frac{r^2}{r^2/4}=4\text{ triangles per square}.$$

**Number of squares.** With 12 triangles, each used once, $12/4=3$ squares can be formed.

Answer: **3 squares, 4 triangles each** (option A).