Multiple choice

What is the area of an equilateral triangle whose inscribed circle has radius R?

  1. 4R2

  2. 4 3 R 2

  3. ( 3 + 2 2 ) R 2

  4. 3 3 R 2
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For an equilateral triangle, the inradius is one-third of the altitude, so the side length is 2sqrt(3)R. Using area = sqrt(3)/4 × side^2 gives 3sqrt(3)R^2, corresponding to option D.

AI explanation

For an equilateral triangle, the inradius R is related to the side s by the formula R = s√3 / 6. Solving for the side gives s = 6R / √3, which simplifies to s = 2R√3. Substitute this into the equilateral triangle area formula, Area = (√3/4)s², to get Area = (√3/4)(4)(3)R², which simplifies to 3√3 R². The area of the triangle is 3√3 R².

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