Find a sphere’s volume and surface area — with the full working shown.
The sphere — the set of points at distance r from a center — is nature’s favorite shape: planets, bubbles, droplets, and ball bearings all minimize surface for their volume by going spherical. Two formulas rule it: V = 4/3πr³ and A = 4πr². Our free Sphere Calculator computes both from the radius, shows each step, and adds the diameter.
Archimedes proved the volume formula — and was so proud he asked for a sphere-in-cylinder carving on his tomb. The surface area’s elegance (exactly four great-circle areas) hints at deep symmetry: of all solids with a given surface, the sphere encloses the most volume.
Enter the radius and the calculator returns volume and surface area to two decimals with the full substitution steps. Whether you are sizing a tank, checking a ball spec, or doing geometry homework, the steps are shown and the math is exact.
Volume V = 4/3 πr³: the cubic dependence means doubling the radius multiplies volume by 8 — the single most important scaling fact about spheres. A 6-inch ball holds 8× the 3-inch ball’s contents, not 2×.
Surface area A = 4πr²: exactly four times the area of a great circle (πr²). Paint, plating, and heat loss all scale with r² — doubling the radius quadruples the surface.
The cylinder connection: a sphere fills exactly 2/3 of its circumscribing cylinder’s volume (cylinder: πr² × 2r = 2πr³; sphere: 4/3πr³; ratio 2/3). Archimedes’ tomb carving commemorates this ratio.
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Watch the units: radius in cm gives volume in cm³ and area in cm². Mixing radius with diameter is the #1 sphere error — halve the diameter first.
Find the volume and surface area of a sphere with radius 3.
The matching digits are coincidence (r = 3 makes 4/3×27 = 36 and 4×9 = 36) — a neat check, not a rule.
A spherical tank has diameter 10 m. Find its capacity.
Half a million liters in a 10-meter ball — the r³ scaling makes spheres the densest way to store volume.
A dealer quotes a spherical propane tank, 10 ft diameter, and the buyer wants capacity in gallons. Radius 5 ft: V = 4/3π×125 = 523.6 ft³. Convert: 523.6 × 7.48052 = 3,917 gallons. The widget gave the cubic feet; the volume converter finishes the job — one calculation chain, no guesswork.
A second scenario: a chocolatier coats truffles (r = 1.5 cm) and needs the chocolate per truffle’s surface: A = 4π×2.25 = 28.27 cm². Times 500 truffles, divided by coverage per liter — the batch order writes itself.
A third: a basketball’s circumference is 29.5 in, so r = 29.5/(2π) = 4.695 in. Volume: 4/3π×103.5 = 433.5 in³ — the air inside, computed from the tape measure.
A fourth: Earth (r ≈ 6,371 km): V = 1.083×10^12 km³, A = 5.101×10^8 km² — the 510 million km² surface every geography class quotes, from one formula.
Physics: planets, stars, and droplets are spherical because gravity and surface tension minimize energy — the sphere’s minimal-surface property is why nature chooses it. Newton’s shell theorem (a sphere gravitates as a point mass) unlocks orbital mechanics.
Chemistry: atoms model as spheres (van der Waals radii); close packing of spheres gives crystal structures; reaction rates scale with droplet surface area (r²) while payload scales with volume (r³) — the r²-vs-r³ race governs everything from sprays to cells.
Biology: cells can’t grow arbitrarily large because volume (demand) outruns surface (supply) — the sphere formulas quantify why cells divide. Alveoli multiply surface area to beat the same math.
Engineering: spherical pressure vessels (LPG spheres, diving bells) distribute stress evenly — no corners to concentrate it. The widget’s volume sizes the vessel; the area sizes the steel.
Sports: balls are spheres (within tolerance): golf, tennis, basketballs all spec diameter, and volume/area follow. Dimples tune airflow over the spherical surface.
Astronomy: stellar volumes, planetary surfaces, and the observable universe’s scale all run on 4/3πr³. Earth’s 510 million km² is 4πr² with r = 6,371 km.
Mathematics: the sphere is the 2-sphere S²; its higher-dimensional cousins (3-sphere volume 2π²r³) generalize the formulas. The isoperimetric inequality crowns the sphere optimal among all shapes.
One pattern worth internalizing: r² vs. r³ scaling decides real outcomes — heat loss vs. heat content, paint vs. capacity, metabolism vs. mass. Whenever size changes, ask which power of r each quantity follows.
Ballooning: a party balloon (r ≈ 15 cm) holds 14 liters; a weather balloon (r ≈ 1 m) holds 4,189 liters — 300× the volume from 6.7× the radius. The r³ scaling is why high-altitude balloons look impossibly large.
Cooking: doubling a meatball’s diameter (2× r) makes 8× the volume but only 4× the browning surface — big meatballs stay juicier because surface-to-volume ratio falls as 3/r. The widget’s V and A outputs quantify the recipe change.
Pharmaceuticals: tablet coating weight scales with surface area (r²) while dose scales with volume (r³) — coating a larger tablet needs proportionally less coating per milligram. Scale-up math is sphere math.
Planetary science: Mars (r = 3,390 km) has 28% of Earth’s surface area but only 15% of its volume — (0.532)² vs. (0.532)³. Surface gravity, atmosphere retention, and cooling history all follow from those two powers.High-altitude balloons: a party balloon at r ≈ 15 cm holds 4/3π(0.15)³ ≈ 0.014 m³ — 14 liters — while a weather balloon at r ≈ 1 m holds 4.19 m³, nearly 300 times the volume from 6.7 times the radius. That r³ scaling dictates the entire mission: the envelope must be enormous at launch because the payload’s weight is fixed while the lifting gas expands as outside pressure drops. Engineers size the envelope by working the sphere formula backward from the target float altitude — given the payload mass and the air density at 30 km, solve 4/3πr³ for r. The widget runs it forward; ballooning runs it in reverse.
Cooking science: doubling a meatball’s diameter multiplies its volume by 8 but its browning surface by only 4 — the surface-to-volume ratio falls as 3/r. Big meatballs stay juicier because proportionally less of their mass sits near the hot exterior, and the same r³-versus-r² race explains why restaurant burgers are smashed flat (maximizing surface) while Salisbury steak stays thick. A chef scaling from 2-cm to 4-cm meatballs needs 8× the mixture but the browning dynamics change per unit surface — the widget’s V and A outputs, divided, give the 3/r ratio that predicts the juiciness shift before the first batch hits the pan.
Comparative planetology: Mars (r = 3,390 km) has (3390/6371)² = 28.3% of Earth’s surface area but (3390/6371)³ = 15.1% of its volume. Surface area governs how fast the planet radiates heat away; volume governs how much heat it stored — so Mars cooled faster, its core solidified, and its magnetic field died, all following from two powers of the radius ratio. The Moon (r = 1,737 km) is more extreme: 7.4% of Earth’s surface, 2.0% of its volume. Every “why is Mars dead?” explanation is the sphere formulas applied to 0.532² versus 0.532³.
Sphere: V = 4/3*pi*r^3, A = 4*pi*r^2. Sphere fills 2/3 of its cylinder. Halve diameter to get radius. Source: Sigma Calculator, Sphere Calculator, https://sigmacalculator.net/math-calculators/sphere-calculator/
Cite this page as: Sphere Calculator, Sigma Calculator, https://sigmacalculator.net/math-calculators/sphere-calculator/
Last updated: October 12, 2026