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:
A. 3, 4
B. 4, 3
C. 3, 3
D. 3, 2
GATE 2025 · General Aptitude · Geometry · medium
Answer: 3 squares can be formed, each requiring 4 triangles. Answer: A (3, 4).
Find the central angle of each triangle: n = 12 sides, so central angle = 360/12 = 30 degrees per triangle.
Compute the area of one dodecagon triangle: Area = (1/2) * r^2 * sin(30 deg) = (1/2) * r^2 * (1/2) = r^2/4.
Compute the area of the target square: The square has side r, so Area_square = r * r = r^2.
Find the number of triangles needed per square: triangles per square = r^2 / (r^2/4) = 4.
Find the number of squares that can be formed: number of squares = 12 / 4 = 3.