Final answer:
The equation C=\u03c0r^2 represents the area of a circle, which when calculated using a calculator with eight-digit output and a radius of 1.2 meters, results in 4.5238934 m^2, but with two significant figures, it is approximated to 4.5 m^2.
Step-by-step explanation:
The equation C=\u03c0r^2 is used to calculate the area of a circle. When given a circle with a radius of 1.2 meters, using \u03c0 approximated to 3.1415927, you would calculate the area as follows: A = \u03c0r^2 = (3.1415927...)\u00d7(1.2 m)^2 = 4.5238934 m^2. However, since our radius was given with only two significant figures, the calculated area should also be represented with two significant figures, resulting in A=4.5 m^2.
The ancient Greeks also employed this method over 2000 years ago, recognizing that \u03c0 (pi) is slightly larger than 3. The area of a circle is indeed smaller than the area of a surrounding square but is larger than half of the square's area. By approximation, the area of a circle within a square could be around three-quarters of the square's area, which leads to the estimate of \u03c0r^2 for the area of the circle enclosed within the square with sides of length 4r.