What is the formula for the surface area of a sphere?

The formula for the surface area of a sphere is derived from the integration of circular cross-sections that make up the sphere. To understand why the formula is 4πr², it's helpful to visualize the nature of a sphere.

A sphere is defined as the set of points in three-dimensional space that are a fixed distance (the radius, r) from a given point (the center). When calculating the surface area, we consider the entire curved surface that encloses this volume.

The factor of 4 in the formula arises from the geometry of the sphere and reflects how the area scales with the square of the radius. In this case, you take the radius, square it to account for the two-dimensional nature of surface area, and then multiply by 4π. The π comes from the relationship of the sphere's circular dimensions to its surface area.

This 4πr² formula can also be confirmed through calculus but is widely accepted as a fundamental property of spheres. This makes it essential knowledge in geometry, particularly when dealing with three-dimensional shapes.