Sphere Volume and Surface Area Calculator
A sphere is a perfectly round three-dimensional solid where every point on the surface is equidistant from the center. Enter the radius (r) below to instantly compute two fundamental properties: - Volume (V = 4/3 · π · r³) — how much space the sphere encloses - Surface Area (A = 4 · π · r²) — the total area of its outer shell Use any unit (cm, m, inches, feet) — just be consistent. These formulas apply everywhere: basketballs, water tanks, planets, soap bubbles.
For a sphere of radius r: Volume = (4/3) × π × r³ and Surface Area = 4 × π × r². Example: r = 5 cm → Volume = 523.6 cm³, Surface Area = 314.2 cm². Doubling the radius multiplies volume by 8 and surface area by 4.
When to use this calculator
- Finding the capacity of a spherical water tank or storage vessel (convert cm³ to liters by dividing by 1,000).
- Estimating the amount of paint or material needed to coat a spherical surface — a dome, ball, or globe.
- Solving geometry homework or standardized test problems (SAT, ACT, AP Calculus) involving spheres.
- Engineering: computing the mass of a solid metal sphere from V × density (e.g., steel ≈ 7.85 g/cm³).
- Science: estimating the volume or surface area of planets, cells, or fluid droplets given their radius.
Worked Example — Basketball
- NBA basketball radius ≈ 4.70 in
- V = (4/3) × π × (4.70)³ = (4/3) × π × 103.82 ≈ 434.9 in³
- A = 4 × π × (4.70)² = 4 × π × 22.09 ≈ 277.6 in²
How it works
2 min readFormulas
The two core formulas for a sphere of radius r were proven by Archimedes (~250 BCE):
Volume: V = (4/3) · π · r³
Surface Area: A = 4 · π · r²
Where:
r = radius (= diameter / 2)
π ≈ 3.14159265358979If you know the diameter (d):
V = (π · d³) / 6
A = π · d²If you know the circumference (C):
r = C / (2π)---
Reference Table — Volume & Surface Area by Radius
| Radius (cm) | Diameter (cm) | Volume (cm³) | Surface Area (cm²) |
|---|---|---|---|
| 1 | 2 | 4.19 | 12.57 |
| 2 | 4 | 33.51 | 50.27 |
| 3 | 6 | 113.10 | 113.10 |
| 4 | 8 | 268.08 | 201.06 |
| 5 | 10 | 523.60 | 314.16 |
| 6 | 12 | 904.78 | 452.39 |
| 7 | 14 | 1,436.76 | 615.75 |
| 8 | 16 | 2,144.66 | 804.25 |
| 9 | 18 | 3,053.63 | 1,017.88 |
| 10 | 20 | 4,188.79 | 1,256.64 |
| 15 | 30 | 14,137.17 | 2,827.43 |
| 20 | 40 | 33,510.32 | 5,026.55 |
| 25 | 50 | 65,449.85 | 7,853.98 |
| 30 | 60 | 113,097.34 | 11,309.73 |
| 50 | 100 | 523,598.78 | 31,415.93 |
> Tip: For r = 3 cm, volume and surface area are numerically equal (both ≈ 113.10) — a curious coincidence, not a general rule.
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Worked Examples
Example 1 — Basketball (r = 4.70 in)
V = (4/3) · π · (4.70)³ ≈ 434.9 in³
A = 4 · π · (4.70)² ≈ 277.6 in²Example 2 — Spherical Water Tank (r = 2 m)
V = (4/3) · π · 8 ≈ 33.51 m³ → 33,510 liters ≈ 8,850 US gallons
A = 4 · π · 4 ≈ 50.27 m²Example 3 — Steel Ball Bearing (r = 1.5 cm, ρ = 7.85 g/cm³)
V = (4/3) · π · (1.5)³ ≈ 14.137 cm³
Mass = 14.137 × 7.85 ≈ 111.0 g
A = 4 · π · (1.5)² ≈ 28.27 cm²---
Common Mistakes
1. Radius vs. diameter confusion. Entering the diameter (e.g., 10 cm) instead of the radius (5 cm) inflates volume by 8× because r is cubed. Always halve the diameter first.
2. Forgetting the 4/3 coefficient. V = πr³ is wrong. The correct formula has a 4/3 factor (≈ 1.3333), and omitting it understates volume by ~25%.
3. Mixing units. If r is in mm but you want cm³, convert first (1 cm = 10 mm). Volume scales by the cube: 1 cm³ = 1,000 mm³.
4. Rounding π too early. π ≈ 3.14 gives ~0.05% error, which compounds when cubing. Use at least 5 decimal places (3.14159) for engineering work.
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Frequently asked questions
What is the formula for the volume of a sphere?
V = (4/3) · π · r³, where r is the radius. For r = 5 cm: V = (4/3) × π × 125 ≈ 523.6 cm³. This formula was proven by Archimedes around 250 BCE using the method of exhaustion.
What is the formula for the surface area of a sphere?
A = 4 · π · r². For r = 5 cm: A = 4 × π × 25 ≈ 314.16 cm². This equals exactly four times the area of a great circle (πr²) — a relationship Archimedes proved in On the Sphere and Cylinder.
How do I calculate sphere volume from the diameter?
Divide the diameter by 2 to get the radius, then apply V = (4/3)πr³. Or use the direct formula V = (π · d³) / 6. For d = 12 cm: V = π × 1728 / 6 ≈ 904.78 cm³.
How do I convert sphere volume from cm³ to liters or gallons?
Use: 1 liter = 1,000 cm³ and 1 US gallon = 3,785.41 cm³. A sphere with r = 20 cm has V ≈ 33,510 cm³ = 33.51 liters ≈ 8.85 US gallons.
Does doubling the radius double the volume?
No — volume scales with r³, so doubling the radius multiplies volume by 2³ = 8. Example: r = 5 cm → V ≈ 523.6 cm³; r = 10 cm → V ≈ 4,188.8 cm³ (8× larger). Surface area scales with r², so doubling r increases it by 4×.
What is the relationship between a sphere and its enclosing cylinder?
A sphere of radius r fits inside a cylinder of radius r and height 2r. The sphere's volume is exactly 2/3 of the cylinder's volume (2πr³). Archimedes considered this ratio his greatest discovery and asked for it to be inscribed on his tomb.
Can I use this calculator for hollow spheres (spherical shells)?
For a hollow sphere with outer radius R and inner radius r: shell volume = (4/3)π(R³ − r³) and surface area = 4π(R² + r²) (both surfaces). Run this calculator twice and subtract the volumes — or add the two surface areas.
What are some real-world examples where sphere formulas are used?
Basketball sizing (NBA diameter ≈ 9.39 in, V ≈ 434.9 in³), spherical water tanks (r = 2 m holds ~33,510 liters), steel ball bearings (r = 1.5 cm, mass ≈ 111 g), planetary science (Earth's mean radius ≈ 6,371 km, V ≈ 1.083 × 10¹² km³), and soap bubbles (minimizing surface area for a given volume is why they are spherical).
Why is π (pi) used in the sphere formulas?
π appears because spheres are inherently circular — every cross-section through the center is a circle of area πr². Volume integrates infinitely many such circles, and surface area is the limiting case of many small circular patches. π ≈ 3.14159265... is irrational and transcendental, so all sphere volumes/areas involving exact dimensions are also irrational.