Four identical circular cardboard pieces, each having a radius of 7 cm, are placed so that each circular piece touches exactly two of the others. What is the area of the region enclosed between the four circular pieces?

Introduction:
This geometry question involves four identical circles arranged in a simple symmetric pattern. Each circle touches two others, suggesting that their centres form a square. The problem asks for the area of the empty region enclosed between the four circles. This type of question is common in aptitude and competitive exams to test a student's understanding of circle geometry, area subtraction, and visualisation of composite figures.

Given Data / Assumptions:

• Each circular cardboard piece is a circle of radius 7 cm. • Four such circles are arranged so that each circle touches exactly two others. • The circles are arranged in a square-like configuration, with centres at the corners of a square. • We need the area of the region enclosed in the middle between the four circles.

Concept / Approach:
When four equal circles are arranged so that each touches two others, their centres lie at the vertices of a square. The side of this square equals the diameter of a circle, because centres of two touching circles are separated by a distance of 2 * radius. The enclosed region is obtained by taking the area of this square and subtracting the areas of the four quarter-circles inside the square. Four quarter-circles of the same radius combine to form one complete circle. Thus, the required area is: area of square minus area of one full circle.

Step-by-Step Solution:
Step 1: Compute the side length of the square formed by the centres. Side of square = diameter of circle = 2 * 7 = 14 cm. Step 2: Compute area of this square. Area of square = side * side = 14 * 14 = 196 sq cm. Step 3: Compute area of one circle of radius 7 cm. Area of circle = pi * r^2 = pi * 7^2 = 49 * pi. Step 4: Use pi = 22/7 for calculation. Area of circle = 49 * (22/7) = 49 * (22 / 7) = 7 * 22 = 154 sq cm. Step 5: Compute the enclosed area. Enclosed area = area of square - area of one circle = 196 - 154 = 42 sq cm.

Verification / Alternative check:
We can visualise the configuration: each circle occupies one corner of the square, and inside the square four quarter-circles carve away space from the corners. These four quarter-circles exactly make one full circle of radius 7 cm. The remaining region must be smaller than the square but clearly more than zero. Numerically, 42 sq cm is significantly less than 196 sq cm but not negligible, which matches our geometric intuition. No other meaningful subtraction pattern emerges from this arrangement, so the method is reliable.

Why Other Options Are Wrong:
Option 12 sq cm: This is too small and would imply most of the square is removed, which is not the case. Option 32 sq cm: Still smaller than the correct area and does not match 196 - 154. Option 52 sq cm: This is larger than the correct leftover area and does not correspond to a simple combination of standard areas. Option 72 sq cm: This would require the circle area to be 196 - 72 = 124, which does not match any simple pi * r^2 value for radius 7.

Common Pitfalls:
Students sometimes mistakenly subtract the areas of four full circles from the square, or they subtract only one quarter-circle instead of four. Another common mistake is to take the side of the square as 7 cm instead of 14 cm by forgetting that the distance between centres of two touching circles is the diameter, not the radius. Also, using an incorrect value of pi or rounding too early can lead to approximate but wrong answers.

Final Answer:
The area of the region enclosed by the four circular cardboard pieces is 42 sq cm.