1. The problem asks why the area of a sphere is given by $4\pi r^2$ and not $\frac{4}{3}\pi r^2$, and how this relates to adding infinite circles. 2. First, let's clarify the formulas: - The volume of a sphere is $\frac{4}{3}\pi r^3$. - The surface area of a sphere is $4\pi r^2$. 3. The question seems to confuse volume and surface area. The $\frac{4}{3}\pi r^3$ formula is for volume, which measures the space inside the sphere. 4. The surface area formula $4\pi r^2$ measures the total area covering the sphere's surface. 5. Regarding "adding infinite circles": - Imagine slicing the sphere into infinitely many thin circular cross-sections (disks). - Each disk has an area $\pi y^2$, where $y$ depends on the slice's position. - Integrating these areas along the radius gives the volume, not the surface area. 6. To find the surface area, we use calculus to sum the areas of tiny curved patches, not flat circles. - The formula comes from integrating the circumference of circles times an infinitesimal arc length. 7. Summary: - $\frac{4}{3}\pi r^3$ is volume (3D space inside). - $4\pi r^2$ is surface area (2D covering). - Adding infinite circles (disks) relates to volume, not surface area. Final answer: The area of a sphere is $4\pi r^2$, not $\frac{4}{3}\pi r^2$, because the latter is part of the volume formula, and surface area is calculated differently using calculus.