An equilateral triangle is inscribed in a circle, with each vertex of the triangle lying on the circumference of the...

1. TRANSLATE the problem information

• Given information:
  • Equilateral triangle inscribed in circle (vertices on circumference)
  • Area of triangle = 432√3 square units
  • Need to find radius of circle

2. INFER the solution strategy

• We need to connect triangle area to circle radius
• Strategy: Find triangle side length first, then use geometry to relate side length to radius
• This requires the area formula for equilateral triangles and geometric properties of inscribed triangles

3. SIMPLIFY to find the side length

Using area formula \(\mathrm{A = \frac{s^2\sqrt{3}}{4}}\):

• \(\mathrm{\frac{s^2\sqrt{3}}{4} = 432\sqrt{3}}\)
• Divide by √3: \(\mathrm{\frac{s^2}{4} = 432}\)
• Multiply by 4: \(\mathrm{s^2 = 1728}\)
• Take square root: \(\mathrm{s = \sqrt{1728} = 24\sqrt{3}}\) (use calculator if needed)

4. INFER the geometric relationship

• For an equilateral triangle inscribed in a circle, we can use the Law of Cosines
• The triangle divides into three congruent triangles from center, each with central angle 120°
• For one triangle: \(\mathrm{s^2 = R^2 + R^2 - 2R^2\cos(120°)}\)

5. SIMPLIFY the geometric relationship

• \(\mathrm{s^2 = 2R^2 - 2R^2(-1/2)}\)
• \(\mathrm{s^2 = 2R^2 + R^2}\)
• \(\mathrm{s^2 = 3R^2}\)
• Therefore: \(\mathrm{s = R\sqrt{3}}\)

6. SIMPLIFY to find the radius

• Since \(\mathrm{s = 24\sqrt{3}}\) and \(\mathrm{s = R\sqrt{3}}\):
• \(\mathrm{24\sqrt{3} = R\sqrt{3}}\)
• \(\mathrm{R = 24}\)

Answer: C. 24

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often try to work directly with the area without establishing the connection between triangle properties and circle radius. They may remember area formulas but fail to see that they need an intermediate step (finding side length) before they can relate to the radius.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about inscribed vs. circumscribed: Some students interpret "inscribed in a circle" as the triangle containing the circle (rather than vertices on circumference), leading them to use incorrect geometric relationships.

This causes them to get stuck early in the problem and resort to guessing.

The Bottom Line:

This problem requires connecting area formulas with geometric relationships for inscribed polygons. Success depends on recognizing that the solution requires two main steps: translating area to side length, then relating side length to circumradius through geometric properties.