Sphere Volume

How to Find the Radius From Sphere Volume (Inverse Calculation)

Step-by-step algebraic isolation to solve for radius when you are given the volume of a sphere: r = ³√(3V / 4π).
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📅 Updated: October 13, 2026
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Reverse Engineering: Solving for Radius

In many practical and standardized testing scenarios, you are given the volume of a sphere and tasked with finding its radius. This requires solving the volume equation for r algebraically.

Algebraic Derivation

  1. Start with: V = (4/3)πr³
  2. Multiply both sides by 3: 3V = 4πr³
  3. Divide both sides by 4π: r³ = (3V) / (4π)
  4. Take the cube root of both sides: r = ³√( (3V) / (4π) )

Worked Reverse Problem

A water tank holds V = 500 m³ of water in a spherical vessel. Find its internal radius.

1. Compute 3V: 3 × 500 = 1500

2. Compute 4π: 4 × 3.14159 = 12.56637

3. Divide: 1500 / 12.56637 = 119.366

4. Cube root: ³√(119.366) ≈ 4.92 meters

Sphere Volume Formula

Sphere Volume Formula

V = (4/3) × π × r³
Where V = Volume • r = Radius • π ≈ 3.14159265

Essential Mathematical Notation

  • V Volume of the sphere — The three-dimensional space enclosed within the spherical surface, measured in cubic units (e.g., cm³, m³, in³, ft³).
  • r Radius of the sphere — The straight-line distance from the exact center to any point on the outer surface (r = d / 2).
  • π Pi (π) — The mathematical circle constant, approximately 3.14159265 (or 22/7).
  • 4/3 Geometric Constant (4/3) — The rational coefficient derived from spherical calculus integration.
  • r³ Radius Cubed (r³) — The radius multiplied by itself three times (r × r × r), measuring 3D volume.
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