In an equilateral triangle with side length 'a', what is the ratio of the circumradius to the inradius?

Correct — A. In an equilateral triangle the circumcentre and the incentre are the same point, the centroid, which lies on every median.

Median (= altitude): h = √3⁄2 × a
The centroid divides it 2 : 1 from the vertex
Circumradius: R = 2⁄3 × h = a⁄√3
Inradius: r = 1⁄3 × h = a⁄(2√3)

Divide: R ÷ r = (a⁄√3) × (2√3⁄a) = 2
Ratio R : r = 2 : 1 → option (a)

Why the others are wrong

• (b)
  3:1 — 3 : 1 is the altitude to the inradius, h : r, since r is one-third of the median. The circumradius is two-thirds of it, so R : r = 2 : 1.
• (c)
  4:1 — 4 : 1 is the ratio of the two circles' areas, πR² : πr² = 2² : 1². The question asks for the radii, a length ratio, which stays 2 : 1.
• (d)
  1:2 — 1 : 2 is the ratio reversed, inradius to circumradius. The question names the circumradius first, and the circle through the vertices is the larger one.

In an equilateral triangle the centroid, circumcentre, incentre and orthocentre coincide, at one point on each median.

The centroid cuts every median in the ratio 2 : 1 from the vertex. The longer part runs to a vertex: that is the circumradius. The shorter part runs to the midpoint of a side, where the incircle touches: that is the inradius.

Key facts

• Equilateral triangle: circumradius R = a⁄√3.
• Equilateral triangle: inradius r = a⁄(2√3).
• The altitude √3⁄2 × a equals R + r = 3r.

Common traps

• Picking 4 : 1 by squaring, which is the ratio of the circles' areas, not their radii.
• Reversing the order and answering inradius : circumradius = 1 : 2.

No directly related past PYQ was found.