The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is A. 1/8 B. 1/6 C. 1/4 D. 1/2

GATE 2021 · General Aptitude · Triangles · medium

Answer: Ratio of incircle area to circumcircle area = 1/4.

1. Write inradius of equilateral triangle: For equilateral triangle with side a: inradius r = a*sqrt(3)/6.
2. Write circumradius of equilateral triangle: For equilateral triangle with side a: circumradius R = a*sqrt(3)/3.
3. Compute r/R: r/R = (1/6) / (1/3) = 1/2. The inradius is half the circumradius.
4. Compute area ratio: Area ratio = (1/2)^2 = 1/4.