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GATE 2025 DA – 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.

A regular dodecagon inscribed in a circle, divided from its centre into 12 triangles numbered 1 to 12. The side of the dodecagon is marked d.
  1. 3; 4
  2. 4; 3
  3. 3; 3
  4. 3; 2

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Correct answer: (A) 3; 4

Explanation

Each triangle has two sides equal to $r$ with an angle of $\frac{360^\circ}{12} = 30^\circ$ between them, so its area is $\frac{1}{2}r^2\sin 30^\circ = \frac{r^2}{4}$. A square of side $r$ has area $r^2$, so it takes $\frac{r^2}{r^2/4} = 4$ triangles. Twelve triangles make $\frac{12}{4} = 3$ squares. That is 3 squares of 4 triangles each.