Calculators

Sphere Surface Area Calculator

Find the total outer surface area of a sphere with radius or diameter inputs and step-by-step formulas.
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📅 Updated: October 13, 2026
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Sphere Surface Area Calculator & Proofs

Calculate the total exterior area covering a sphere using Archimedes' theorem A = 4πr². Discover how sphere surface area relates to four times the area of its great circle.

Sphere Surface Area Formula
A = 4 × π × r²
  • A = Total surface area in square units (cm², m², in²)
  • r = Radius of the sphere
  • π = Mathematical constant Pi (≈ 3.14159)

Step-by-Step Solution

  1. Find radius: If diameter is given, divide by 2 (r = d / 2).
  2. Square radius: Multiply radius by itself (r²).
  3. Multiply by 4π: Multiply r² by 4 and then by π.

Worked Demonstration

For a sphere with radius r = 6 cm:
A = 4 × π × 6² = 4 × π × 36 = 144π ≈ 452.39 cm².

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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