The correct answer is option (C):
48 sq.m
Let's break down this geometry problem step-by-step.
First, visualize the situation. We have a square, and on each side of the square, a circle is drawn such that the side of the square is the diameter of the circle. The circles touch each other at the midpoints of the square's sides.
The key to solving this is to find the area of the square and the combined area of the four circles, then subtract the area of the circles from the area of the square.
The square's side length is given as 15 meters. Therefore, the area of the square is side * side = 15 m * 15 m = 225 sq.m.
Now, consider one of the circles. The diameter of each circle is equal to the side of the square, which is 15 meters. Therefore, the radius of each circle is half the diameter, which is 15 m / 2 = 7.5 m.
The area of a single circle is calculated using the formula pi * radius^2. So, the area of one circle is approximately 3.14 * (7.5 m)^2 = 3.14 * 56.25 sq.m = 176.625 sq.m.
Since there are four identical circles, the total area of the four circles is 4 * 176.625 sq.m = 706.5 sq.m. However, this is incorrect. The four circles, placed as described, overlap. Looking at the situation, the four circles fill the square. Each quarter of each circle overlaps with the neighboring quarters of the adjacent circles, filling the square.
The question asks for the area of the square *not* covered by the circles. In this particular setup, the circles cover the whole square, but since there is an overlap, the circles are not completely filling the whole area. Consider the four quarter circles placed on each side. Together, they create one full circle inside the square. However, there will be four curved portions of the square which are not part of the circle. So, the area inside the circles must be deducted. The question is slightly tricky since it is not asking for the area of the circles. It's asking for the area of the square minus the area covered by the circles.
We can analyze this in a different way. Imagine that we've drawn the four semicircles on the sides of the square. The parts of the circle outside the square will cancel each other out. This means we must subtract the quarter circles (which form the inner full circle) of each side.
The diameter of each circle is 15 m, which is the same as the length of the square. The circles cover the total area of the square, but as we calculate the area of the four circles, we see that they only cover parts of the square as well as parts of each other. The area of the square not covered by the circles is the area that remains. This area is the square area - circle area. That is, 225 sq.m - (3.14 * 7.5^2) = 225 - 176.625 = ~48 sq.m.
Therefore, the area of the square not covered by the four circles is approximately 48 sq.m.