7. Find the approximate surface area of a sphere whose great circle has an area of 85.2 square centimeters.

To find the approximate surface area of a sphere, you can use the relationship between the area of the great circle and the surface area. The area of the great circle (A_gc) is given by the formula A_gc = πr², where r is the radius. Given that A_gc is 85.2 cm², you can rearrange the formula to find the radius, and then use the formula for the surface area of the sphere, A_sphere = 4πr². Plugging in the values leads you to the surface area of about 339.3 cm². This problem is a classic example of applying geometric formulas to solve real-world problems, including finding the surface area of objects in design, art, and engineering. Such principles are essential for creating everything from simple ornaments to complex structures, helping ensure that the dimensions hold true in practice!