Volume of a Sphere
The volume of a sphere measures the total three-dimensional space enclosed within its surface. The formula was first derived by Archimedes, who showed that a sphere's volume is exactly two-thirds the volume of its circumscribing cylinder.
Formulas
From Radius
The standard formula using the sphere's radius.
V = (4/3) x pi x r³
From Diameter
Substitute r = d/2 into the standard formula.
V = (pi x d³) / 6
From Circumference
Substitute r = C / 2pi into the standard formula.
V = C³ / (6 x pi²)
From Surface Area
Derive r from SA, then compute volume.
V = (SA^(3/2)) / (6 x sqrt(pi))
Derivation of the Formula
The sphere volume formula can be derived using calculus by integrating the area of circular cross-sections (disks) from the bottom to the top of the sphere. Each disk at height y has radius sqrt(r² - y²), and integrating pi(r² - y²) from -r to r yields (4/3)pi r³.
Real-World Examples
- Basketball: A regulation basketball has a diameter of about 24 cm, giving a volume of approximately 7,238 cm³.
- Earth: With a radius of 6,371 km, Earth's volume is about 1.083 x 10¹² km³.
- Water balloon: A 10 cm diameter balloon holds about 524 mL of water.