In an equilateral triangle, the radius of the circumscribed circle (circumcircle) is 14 cm. What is the radius of the inscribed circle (incircle) in centimetres?

Introduction / Context:
This geometry question is about the relationship between the circumradius and inradius of an equilateral triangle. An equilateral triangle has all sides equal and all internal angles equal to 60 degrees. For such a triangle, the circumradius and inradius are directly related by simple formulas, which allows us to find one when the other is known.

Given Data / Assumptions:

• Triangle is equilateral.
• Circumradius R = 14 cm.
• Side length of equilateral triangle is denoted by a.
• Standard formulas: R = a / (sqrt(3)) and r = a / (2 * sqrt(3)), where r is inradius.
• Alternatively, r = R / 2 for an equilateral triangle.

Concept / Approach:
For an equilateral triangle, the circumradius R and inradius r have a fixed ratio. Since R = a / sqrt(3) and r = a / (2 * sqrt(3)), dividing r by R gives r / R = 1/2. Thus r = R / 2. With R known as 14 cm, we can directly find r by halving R. This approach avoids having to compute the side length explicitly, though we can still do so as a check.

Step-by-Step Solution:
Given circumradius R = 14 cm.For an equilateral triangle, r = R / 2.So inradius r = 14 / 2 = 7 cm.Therefore, the radius of the inscribed circle is 7 cm.

Verification / Alternative check:
We can verify by computing the side length. From R = a / sqrt(3), we have a = R * sqrt(3) = 14 * sqrt(3). Then r = a / (2 * sqrt(3)) = (14 * sqrt(3)) / (2 * sqrt(3)) = 14 / 2 = 7 cm. This alternative method matches the earlier result and confirms that r = 7 cm is correct.

Why Other Options Are Wrong:
14 cm would make inradius equal to circumradius, which is not true for an equilateral triangle. 28 cm and 35 cm are larger than the circumradius and cannot be the inradius. 21 cm would imply r = (3/2) * R, which contradicts the known geometric relation. Only 7 cm correctly reflects the relationship between inradius and circumradius in an equilateral triangle.

Common Pitfalls:
A frequent mistake is to confuse inradius and circumradius formulas, or to apply formulas that belong to general triangles instead of the special equilateral case. Some learners attempt to compute side length and mis-handle the square roots. Remembering the simple relation r = R / 2 for equilateral triangles is a fast and reliable way to answer such questions. Always ensure that inradius is smaller than circumradius, which helps catch obvious errors.

Final Answer:
7 cm