Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure. The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is
A. 2:3 B. 3:4 C. 4:5 D. 5:6
GATE 2021 · General Aptitude · Triangles · medium
Answer: Ratio of hexagon area to original triangle area = 2:3.
Set up side lengths: Let original equilateral triangle have side 3a. Each corner cut removes an equilateral triangle of side a (trisecting each side gives the regular hexagon property).
Compute original triangle area: A_original = (sqrt(3)/4)*(3a)^2 = (9*sqrt(3)/4)*a^2
Compute area of one corner triangle: A_corner = (sqrt(3)/4)*a^2. Three corners: 3 * (sqrt(3)/4)*a^2 = (3*sqrt(3)/4)*a^2.
Compute hexagon area and ratio: A_hex = (9*sqrt(3)/4)*a^2 - (3*sqrt(3)/4)*a^2 = (6*sqrt(3)/4)*a^2. Ratio = A_hex / A_original = (6*sqrt(3)/4)*a^2 / (9*sqrt(3)/4)*a^2 = 6/9 = 2/3.