In an equilateral triangle with side length 12 cm, what is the numerical difference (in cm) between its circumradius and its inradius?

Introduction / Context:
In this geometry question we work with an equilateral triangle and the special radii associated with it, namely the circumradius and the inradius. The circumradius is the radius of the circle that passes through all three vertices, while the inradius is the radius of the circle that touches all three sides. For an equilateral triangle there are standard formulas that connect the side length to these radii. Once we know both radii we can find the required difference between them.

Given Data / Assumptions:
- The triangle is equilateral, so all sides are equal and all angles are 60 degrees. - Each side of the triangle has length 12 cm. - We need the difference circumradius − inradius expressed in centimetres. - Standard formulas for an equilateral triangle are applicable.

Concept / Approach:
For an equilateral triangle of side a, the circumradius R and inradius r satisfy simple relationships with a. The formulas are R = a / √3 and r = a / (2√3). These results come from basic geometry using centroid, circumcenter and incenter properties, or from trigonometry in a 30 60 90 triangle. Once we substitute a = 12, we compute R and r separately and then take the difference R − r. Because both radii are positive and R is larger, the difference is simply R minus r with no absolute value issues.

Step-by-Step Solution:
Step 1: Let the side length be a = 12 cm. Step 2: Use the circumradius formula R = a / √3, so R = 12 / √3. Step 3: Use the inradius formula r = a / (2√3), so r = 12 / (2√3). Step 4: Simplify r = 12 / (2√3) = 6 / √3. Step 5: Compute the required difference R − r = (12 / √3) − (6 / √3). Step 6: Since denominators match, subtract numerators: R − r = (12 − 6) / √3 = 6 / √3. Step 7: Rationalise 6 / √3 if desired: 6 / √3 = (6√3) / 3 = 2√3. Step 8: Therefore the difference between circumradius and inradius is 2√3 cm.

Verification / Alternative check:
We can also observe that for an equilateral triangle r = a / (2√3) and R = a / √3. This means R is exactly twice r. So R − r = r. For a = 12 cm we have r = 6 / √3 = 2√3, which matches the value we have already computed. This quick reasoning confirms that our algebraic steps are consistent and that the final difference is correct.

Why Other Options Are Wrong:
Option 2√2 would require the difference to involve a factor of √2, which does not appear in the standard formulas for equilateral triangle radii. Option 3√2 again introduces √2 and also the factor 3, neither of which arises when we simplify 6 / √3. Option 3√3 would mean the difference is larger than the inradius itself, which contradicts the relationship R − r = r for an equilateral triangle.

Common Pitfalls:
Many learners confuse the formulas for circumradius and inradius or mix them with those for right triangles. Another frequent mistake is to rationalise incorrectly when dealing with denominators containing square roots. Some students also attempt to use area formulas first, which is longer and more error prone here. It is safer to memorise and apply the direct formulas R = a / √3 and r = a / (2√3) specifically for equilateral triangles.

Final Answer:
The difference between the circumradius and the inradius of the triangle is 2√3 centimetres.