General | Math & Conversions Sphere Volume Calculator Sphere Volume Calculator ($V = \frac{4}{3} \pi r^3$) Calculate the volume ($V$), surface area ($A$), and circumference ($C$) of a sphere. Enter either the **Radius ($r$)** or the **Diameter ($d$)** below, and the tool will solve for all other properties. Enter EITHER Radius ($r$) or Diameter ($d$) Radius ($r$) Diameter ($d$) Calculate Sphere Key Results Radius ($r$) Surface Area ($A$) Circumference ($C$) Volume ($V$) Step-by-Step Solution Formulas for Sphere Volume and Area The volume and surface area formulas for a sphere, first rigorously derived by Archimedes, demonstrate a simple and elegant relationship with the radius, $r$. 1. Volume ($V$) The volume of the sphere is determined by cubing the radius: $$V = \frac{4}{3} \pi r^3$$ Historically, Archimedes showed that the volume of a sphere is two-thirds the volume of the smallest cylinder that can enclose it. 2. Surface Area ($A$) The surface area formula is exactly four times the area of the sphere's great circle ($\pi r^2$, the cross-section passing through the center): $$A = 4 \pi r^2$$ Key Sphere Terminology **Radius ($r$):** The distance from the center to any point on the surface. **Diameter ($d$):** The distance across the sphere passing through the center ($d = 2r$). **Great Circle:** The largest circle that can be drawn on the surface of the sphere. Its circumference is $C = 2\pi r$. Frequently Asked Questions (FAQ) What is the formula for the volume of a sphere? The volume ($V$) of a sphere is given by the formula: $V = \frac{4}{3} \pi r^3$, where $r$ is the radius. What is the formula for the surface area of a sphere? The surface area ($A$) of a sphere is given by the formula: $A = 4 \pi r^2$, which is four times the area of its great circle. How does a sphere's volume relate to a cylinder's volume? The volume of a sphere is $\frac{2}{3}$ the volume of a cylinder that perfectly encloses it (a circumscribed cylinder). This relationship was a source of great pride for Archimedes, who first derived these formulas. Is a sphere a polyhedron? No. A polyhedron is a three-dimensional shape with flat polygonal faces (like a cube or pyramid). A sphere has a continuously curved surface, making it a curved geometric shape, not a polyhedron. Sphere Key Formulas Volume $$V = \frac{4}{3} \pi r^3$$ Surface Area $$A = 4 \pi r^2$$ Circumference $$C = 2 \pi r$$ Related Solid Geometry Tools Cylinder Volume Calculator Cone Volume Calculator Volume Calculator (General) Surface Area Calculator (General) Area Calculator (Circle)