Sphere area and volume: doubling the radius changes them differently
Separate square and cubic units, convert diameter to radius, and compare how a sphere grows.
Method
A sphere’s radius runs from its centre to the surface; its diameter crosses the centre from one side to the other. A diameter of 4 cm therefore means a radius of 2 cm. Surface area measures the outer covering in square units, while volume measures the enclosed space in cubic units. These answer different questions and cannot be added together.
For a complete sphere, surface area is 4π times radius squared, and volume is four-thirds π times radius cubed. Use the same length unit throughout. Multiplying radius by a factor k multiplies area by k² and volume by k³. Doubling radius therefore gives four times the area and eight times the volume, not twice each result.
- Read the radiusd = 4 cm → r = 2 cm
- Calculate surface areaA = 4πr² → 16π cm²
- Calculate volumeV = 4πr³/3 → (32π ÷ 3) cm³
Separate square and cubic units, convert diameter to radius, and compare how a sphere grows.
Worked example
Take an ideal sphere with radius 2 cm. Its area is 16π cm², approximately 50.27 cm²; its volume is 32π/3 cm³, approximately 33.51 cm³. At radius 4 cm, area becomes 64π cm², approximately 201.06 cm², and volume becomes 256π/3 cm³, approximately 268.08 cm³. The example compares radii, not diameters, and rounding is applied only to the displayed answers.
In Sphere volume and surface area, enter radius 2, then repeat with 4. The calculator lists volume first and surface area second: keep their cubic and square units attached. If you measured diameter, divide it by two before entering it. A circle calculation with the same radius gives the area of a flat central disk, which is one-quarter of the whole sphere’s surface area.
r = 2 cm: A = 16π cm² ≈ 50.27 cm²; V = (32π ÷ 3) cm³ ≈ 33.51 cm³
r = 4 cm: A = 64π cm² ≈ 201.06 cm²; V = (256π ÷ 3) cm³ ≈ 268.08 cm³
r₂ ÷ r₁ = 2; A₂ ÷ A₁ = 4; V₂ ÷ V₁ = 8
- Radius factor
- 2 ×2 cm → 4 cm
- Surface area factor
- 4 ×16π cm² → 64π cm²
- Volume factor
- 8 ×(32π ÷ 3) cm³ → (256π ÷ 3) cm³
Radius changes from 2 cm to 4 cm. Each bar compares one quantity with its own original value; the bars are dimensionless multipliers, not mixed area and volume measurements.
Checks and limits
These formulas assume a perfect, complete sphere. An outside measurement of a hollow ball describes its outside geometry, not usable internal capacity; wall thickness changes the inside radius. A hemisphere or a cut spherical segment needs a different surface calculation. Measurement uncertainty affects the result, and extra displayed digits do not make the original radius more accurate. Check the shape and the quantity needed before applying either number.
Related calculators
Source: OpenStax — Prealgebra 2e