To calculate the surface area of a sphere, which of the following formulas would you use?

The formula for calculating the surface area of a sphere is derived from the relationship between the radius of the sphere and the area it encompasses. The correct formula is represented as 4 times pi times the radius squared, which is noted as 4(pi)r^2.

This formulation comes from integral calculus, where the surface area of a sphere is found by integrating the infinitesimal circular areas that make up the surface. The factor of 4 arises because a sphere has essentially four times the area of great circles (which are the largest circles that can be drawn on the sphere). By using the radius squared (r^2) in the formula, you account for all dimensions of the sphere as it expands in a three-dimensional space.

The other options do not correctly represent the relationship between the radius and the surface area of a sphere. For example, 2(pi)r^2 represents the area of a circle, not a sphere, while (1/3)4(pi)r^2 does not apply to the sphere at all. Lastly, 4(pi)r incorrectly suggests a linear measurement, indicating only the circumference of a great circle rather than the total surface area. Thus, the formula 4(pi)r^2