A circle is circumscribed about an equilateral triangle, and the same…
2025
A circle is circumscribed about an equilateral triangle, and the same equilateral triangle is circumscribed about another circle. What is the ratio of the perimeter of the circumscribed circle to that of the inscribed circle?
- A.
3 : 4
- B.
1 : 2
- C.
2 : 1
- D.
4 : 3
Attempted by 11 students.
Show answer & explanation
Correct answer: C
Concept: In any equilateral triangle, the circumcenter and incenter coincide at the centroid, which fixes a constant relationship between the circumradius (R) and the inradius (r): R = 2r. This ratio never depends on the triangle's size — it holds for every equilateral triangle.
Let the side of the equilateral triangle be a.
Circumradius of the equilateral triangle: R = a/√3.
Inradius of the equilateral triangle: r = a/(2√3).
The perimeter of a circle is 2π × radius, so the ratio of the perimeter of the circumscribed circle to that of the inscribed circle equals the ratio of their radii, R : r.
Substituting: R/r = (a/√3) ÷ (a/(2√3)) = 2, so the ratio of perimeters is 2 : 1.
Cross-check: Taking a concrete side, say a = 2√3, gives R = 2 and r = 1 — the same 2 : 1 ratio — confirming the relationship R = 2r independently of the value chosen for a.