An equilateral triangle of side 6 cm is inscribed in a circle. What is the radius of the circle (the circumradius) in centimetres?

Introduction / Context:
This question deals with the circumradius of an equilateral triangle. When an equilateral triangle is inscribed in a circle, the circle is its circumcircle and the radius is called the circumradius. There is a well known relationship between the side length of an equilateral triangle and its circumradius, which lets you solve such problems very quickly.

Given Data / Assumptions:

• The triangle is equilateral.
• Side length a = 6 cm.
• The triangle is inscribed in a circle, so the circle is the circumcircle of the triangle.
• We need to find the radius R of this circle.
• Standard formula for an equilateral triangle: R = a / sqrt(3).

Concept / Approach:
The circumradius of a general triangle can be expressed in terms of sides and angles, but for an equilateral triangle the formula simplifies greatly. Using geometry or trigonometry, one can show that R = a / sqrt(3). Here, we substitute a = 6 and then rationalise or simplify the result. Knowing this special case formula is very helpful in many geometry problems involving equilateral triangles and circles.

Step-by-Step Solution:
Side of the equilateral triangle a = 6 cm. For an equilateral triangle, circumradius R = a / sqrt(3). So R = 6 / sqrt(3). Multiply numerator and denominator by sqrt(3) to simplify: R = (6 * sqrt(3)) / 3. Simplify 6 / 3 to get R = 2 * sqrt(3) cm.

Verification / Alternative check:
You can verify the relationship using the fact that the centroid, circumcentre and incentre of an equilateral triangle coincide. Drawing radii from the centre to each vertex divides the triangle into three congruent smaller triangles. Working with right triangles formed by dropping an altitude and using basic trigonometric ratios for 30 and 60 degrees also leads to the same circumradius formula, confirming that R = 2 * sqrt(3) cm when a = 6 cm.

Why Other Options Are Wrong:
3√2 cm and 4√3 cm: These are larger than the correct radius and would not fit a side length of only 6 cm. √3 cm: This is half the correct value and corresponds to a radius that is too small. 3 cm: This would correspond to R = a / 2, which is not the circumradius for an equilateral triangle.

Common Pitfalls:
Mixing up circumradius and inradius formulas. The inradius r for an equilateral triangle is a * sqrt(3) / 6, which is not the same as R. Using a generic triangle formula without exploiting the equilateral symmetry. Not rationalising the denominator correctly when dividing by sqrt(3).

Final Answer:
The radius of the circle is 2√3 cm