An equilateral triangle has side length 8 cm. A circumcircle is drawn around the triangle, and an incircle is drawn inside it, touching all three sides. Using π = 22/7, what is the area (in sq.cm) of the region lying between the circumcircle and the incircle of the triangle?

Introduction / Context:
This problem involves an equilateral triangle and the two special circles associated with it: the circumcircle (passing through all vertices) and the incircle (tangent to all three sides). The question asks for the area of the ring-shaped region between these two circles. To solve this, we use known formulas for the circumradius and inradius of an equilateral triangle in terms of its side, compute the areas of both circles, and subtract the incircle area from the circumcircle area.

Given Data / Assumptions:

• Side length of the equilateral triangle, a = 8 cm.
• Use π = 22/7.
• Circumcircle radius is denoted by R.
• Incircle radius is denoted by r.
• We need area of the region between the two circles, that is πR^2 − πr^2.

Concept / Approach:
For an equilateral triangle of side a, the circumradius R and inradius r have standard formulas: R = a / √3 and r = a / (2√3). Once we compute R and r, the area of the circumcircle is πR^2 and the area of the incircle is πr^2. The required area is the difference π(R^2 − r^2). After simplifying R^2 − r^2 symbolically, we substitute the side length and then use π = 22/7 to get a numerical mixed fraction.

Step-by-Step Solution:
Given side a = 8 cm for the equilateral triangle.Circumradius R of an equilateral triangle: R = a / √3.Thus R = 8 / √3.Inradius r of an equilateral triangle: r = a / (2√3).Thus r = 8 / (2√3) = 4 / √3.Compute R^2: R^2 = (8 / √3)^2 = 64 / 3.Compute r^2: r^2 = (4 / √3)^2 = 16 / 3.Difference R^2 − r^2 = 64 / 3 − 16 / 3 = 48 / 3 = 16.Area between circumcircle and incircle = π(R^2 − r^2) = π * 16.Using π = 22 / 7, area = 16 * (22 / 7) = 352 / 7.Convert 352 / 7 to a mixed fraction: 7 * 50 = 350, remainder 2, so 352 / 7 = 50 2/7 sq.cm.

Verification / Alternative check:
We can also compute the areas directly as a check. Circumcircle area = πR^2 = (22 / 7) * (64 / 3) and incircle area = πr^2 = (22 / 7) * (16 / 3). Their difference is (22 / 7) * (64 / 3 − 16 / 3) = (22 / 7) * (48 / 3) = (22 / 7) * 16 = 352 / 7. This reconfirms the simplified result, showing that the intermediate steps are consistent and produce the same final area for the ring-shaped region.

Why Other Options Are Wrong:
The value 50 1/7 sq.cm would correspond to an area of 351 / 7, which is slightly less than the correct 352 / 7. Values 75 1/7 sq.cm and 75 2/7 sq.cm are substantially larger and do not match the derived difference of 16π. The value 40 sq.cm ignores the π factor and is not compatible with π = 22 / 7. Only 50 2/7 sq.cm matches the exact computation based on the given triangle side and circle formulas.

Common Pitfalls:
Typical errors include using incorrect formulas for R or r, such as mixing them up or forgetting the √3 in the denominators. Some students may subtract the radii instead of subtracting the squared radii in the area difference formula. Others may approximate π incorrectly or mishandle the conversion of an improper fraction to a mixed fraction. Carefully applying the correct radius formulas and performing arithmetic step by step avoids these mistakes.

Final Answer:
The area of the region between the circumcircle and the incircle of the equilateral triangle is 50 2/7 sq.cm.