AtlasCalcsUtility Calculation Suite
2026-08-07 11 min

Volume Calculator

Calculate volumes of spheres, cones, cubes, cylinders, tanks, capsules, caps, frustums, ellipsoids, pyramids, and tubes with automatic unit conversion.

Calculator

Shape & Unit

Shape
Unit

V = 4/3 · π · r³ — A perfectly round solid where every surface point is equidistant from the center.

Dimensions

Radius (r)
Results update automatically
Results update live as you type
Status:SphereV = 523.5988 m³523,598.7756 L

Volume Summary

ShapeSphere
Cubic meters523.5988
Liters523,598.7756
Gallons (US)138,320.1357
Cubic feet18,490.7337

All Unit Conversions

UnitValue
Cubic Meters523.5988
Liters523,598.7756
Milliliters523,598,775.5983
Gallons (US)138,320.1357
Quarts (US)553,281.5902
Pints (US)1,106,563.1804
Cups (US)2,213,121.1248
Fluid Ounces (US)17,704,968.9981
Tablespoons (US)35,409,937.9962
Teaspoons (US)106,229,813.9885
Cubic Feet18,490.7337
Cubic Inches31,951,934.6025
Cubic Yards684.841

Results

Optimal

Volume (m³)

523.5988

Sphere via V = 4/3 · π · r³

Liters

523,598.7756

m³ × 1000

Gallons (US)

138,320.1357

m³ × 264.172

Cubic Feet

18,490.7337

m³ × 35.3147

Cubic Inches

31,951,934.6025

m³ × 61023.7

Milliliters

523,598,775.5983

m³ × 1,000,000

Shape

Sphere

V = 4/3 · π · r³

Input Unit

meters

1 meters = 1 m

Closed-form solid geometry normalized to meters before conversion. Ideal-shape net volume — add waste for pourables and confirm irregular vessels by displacement.

Visualizations

Volume in Common Units

523,599Liters
138,320Gallons
18,491Cu Ft
31,951,935Cu In

Fill vs 1000 L (1 m³)

Liters523,599 L

Volume vs 1 m³ Reference

Volume523.6

Formula

Every path converts inputs to meters, applies the closed-form solid formula in m³, then multiplies by fixed unit factors for liters, gallons, feet, and inches. Caps accept any two of r/R/h and derive the third geometrically.

Sphere 4/3·π·r³ · Cylinder π·r²·h · Cone 1/3·π·r²·h · Box l·w·h

Alternative Form

Cap 1/3·π·h²(3R−h) · Frustum 1/3·π·h(r²+rR+R²) · Ellipsoid 4/3·π·abc · Pyramid 1/3·a²h · Tube π(D²−d²)/4·L

Variables

SymbolNameUnitDescription
r, RRadiilengthSmall and large radii (cap, frustum)
hHeight / riselengthAxial height or cap rise
l, w, aLinear edgeslengthBox sides and cube/pyramid edges
D, dTube diameterslengthOuter D must exceed inner d
VVolumem³Converted to 13 output units
fUnit factorm per unite.g. 1 ft = 0.3048 m

Explanation

Volume is length cubed, so consistent units matter more than the formula. Converting first to meters isolates unit risk; the geometry afterwards is exact for ideal solids.

Derivation

Sphere integrates discs π(R²−x²); cylinder is base·height; cone/pyramid are 1/3 of their prism; frustum subtracts similar cones; cap integrates a circular segment; tube subtracts inner from outer cylinder. All reduce to the listed closed forms.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Select the sphere shape and its formula.

V = 4/3 · π · r³
Shape:Sphere
Intermediate: A perfectly round solid where every surface point is equidistant from the center.
Result: formula locked: V = 4/3 · π · r³
Step 2

Convert every dimension to meters with the unit factor.

meters = entered × factor
Unit:meters
Factor:1 meters = 1 m
Intermediate: Radius (r): 5 meters = 5.000000 m
Result: all inputs share one SI baseline
Step 3

Apply the shape formula in cubic meters.

V = 4/3 · π · r³
Volume m³:523.5988
Intermediate: V = 4/3 · π · r³ → 523.5988 m³
Result: V = 523.5988 m³
Step 4

Convert the cubic-meter result into practical units.

L=m³×1000; gal=m³×264.172; ft³=m³×35.3147
Liters:523,598.7756
Gallons (US):138,320.1357
Cubic Feet:18,490.7337
Intermediate: one computation, thirteen label conversions
Result: 523,598.7756 L ≈ 138,320.1357 gal
Step 5

Sanity-check scale and shape choice.

Volume scales with length³
Shape:Sphere
Intermediate: doubling every side multiplies volume 8×
Result: compare against a known reference (bucket ≈ 10 L, tank ≈ m³)
Step 6

Use the result for ordering, dosing, or costing.

material × (1 + waste %)
Order with 10% waste:575.9587 m³
Intermediate: add 5-10% for concrete, soil, and fill
Result: rounded order quantity ready
Final Answer
523.5988

Worked Examples

See how the calculator works with different scenarios.

Scenario

A spherical tank with 5-meter radius.

Inputs

Shape:Sphere
Radius:5 m

Calculation

V = 4/3·π·5³ = 4/3·π·125 = 523.5988 m³ ×1000 = 523,598.78 L ×264.172 = 138,319.49 gal

Result

523.5988 m³ · 523,598.78 L · 138,319.49 gal (US)

Interpretation

One radius drives a cubic result: doubling to 10 m would octuple volume to ~4,188.79 m³. The tank holds over half a million liters.

Understanding Your Result

Your Result: Volume (m³): 523.5988

Typical range: Buckets ~10 L, bathtubs ~300 L, backyard tanks 1-10 m³, pools 20-100 m³, concrete slabs 1-50 m³. Tube walls are liters; caps and frustums span both.

What This Means

The headline volume in m³ plus liters, gallons, cubic feet/inches, and milliliters all describe the same solid. Shape and input unit echo how the number was built.

Why It Matters

Ordering concrete, water, soil, or resin by the wrong unit or the wrong solid wastes money and trips. One SI computation with thirteen conversions removes both risks.

Professional Interpretation

Treat results as ideal-geometry net volumes. Add 5-10% waste for concrete/soil, subtract wall thickness for capacity, and confirm displacement for irregular vessels.

Recommendations

  • •Confirm radius vs diameter before typing
  • •Keep every dimension in the selected unit
  • •Match capsule/frustum/cap to the true profile
  • •Add waste factors after the net computation
  • •Use liters/gallons for fluids, m³/yd³ for bulk
  • •Re-measure the smallest dimension — it cubes the error

Educational Content

Introduction

Volume totals the unit cubes inside a solid. This tool covers eleven closed forms — from spheres and boxes to caps, frustums, and tubes — converting all inputs to meters first, then fanning out to thirteen output units.

Concept Explanation

Each formula multiplies three lengths (length³). Spheres/cylinders/cones derive from circular cross-sections; boxes from edge products; pyramids/cones take one-third of their prism; frustums and caps difference those primitives.

Applications

▸Dosing pools, aquariums, and chemical tanks
▸Ordering concrete, gravel, soil, and mulch
▸Sizing fuel, water, and HVAC vessels
▸Estimating shipping cubes and container fill
▸Computing pipe-wall material and bore capacity
▸Formulating medicine and lab reagents by volume

Benefits

  • •Eleven solids in one converter, no retyping
  • •Eleven input length units with SI normalization
  • •Thirteen output units from one computation
  • •Cap solver derives the missing third dimension
  • •Live 3D-style diagram labels what to measure

Limitations

  • •Ideal geometry only — dents and tapers need displacement
  • •Uniform units required across all inputs
  • •Display rounds to 4/2 decimals
  • •Wall thickness must be modeled explicitly (tube)

Best Practices

  • •Measure inside dimensions for capacity, outside for material
  • •Photograph the profile to justify frustum vs cylinder
  • •Convert once via the unit menu, never by hand mid-form
  • •Add documented waste after net volume
  • •Cross-check sphere/cylinder inversions (r from V) on critical jobs

Common Misconceptions

Myth: Doubling dimensions doubles volume.
Truth: Volume cubes: doubling every side multiplies volume 8×. A 10 m sphere holds 8× a 5 m sphere.
Myth: A capsule is just a cylinder.
Truth: Ignoring the two hemispherical ends drops a full sphere (4/3·π·r³) from the total.
Myth: Gallons and cubic feet convert by 10s.
Truth: 1 ft³ = 7.48052 US gal and 1 gal = 0.133681 ft³ — non-decimal by definition.
Myth: Tube volume is what the pipe holds.
Truth: The tube shape reports wall material (outer minus inner). Bore capacity uses the inner diameter as a plain cylinder.

Practical Tips

  • •Halve diameters to radii before judging sphere inputs
  • •Keep all dimensions in the selected unit
  • •Pick frustum for tapered pots, cone only for true points
  • •Give caps any two of r / R / h
  • •Enforce D > d on tubes or the wall is zero
  • •Add 5-10% waste after net for pourable materials
  • •Use displacement for lumpy, dented, or organic shapes

Common Mistakes

Entering diameter as radius
Halve the tape reading first. A 10 cm diameter is r=5 cm; cubing the full 10 overstates spheres 8×.
Mixing feet and inches in one form
Convert everything to the selected unit first. The solver multiplies by one factor only.
Using cylinder for a frustum tank
Tapered vessels need the frustum form with both radii — cylinders over/understate by the taper gap.
Expecting tube capacity from wall math
For what a pipe holds, run a cylinder on the inner diameter; the tube entry is wall material.
Ordering exact net without waste
Multiply net by 1.05-1.10 for concrete, soil, and mulch to cover spillage and compaction.

Assumptions

This calculator makes the following assumptions to provide estimates:

Ideal solids: Shapes are mathematically perfect with uniform walls.
Single unit: All dimensions share the selected length unit.
Positive inputs: Zero, negative, or blank entries count as missing.
Cap consistency: Two cap inputs must satisfy r² = h(2R−h) within tolerance.
US liquid units: Gallons, quarts, pints, and cups are US customary.

Limitations

Be aware of these limitations when using this calculator:

No irregular bodies: Lumpy or dented vessels need displacement or 3D scanning.
Thin-wall idealization: Capacity vs material split is exact only for uniform thickness.
Display rounding: Outputs round for reading; chain calculations from m³.
Static geometry: Thermal expansion and deformation are not modeled.

Frequently Asked Questions

Answers to common questions about this calculator.

What is volume?
Volume is the three-dimensional space an object or substance occupies, measured in cubic units such as cubic meters, liters, or gallons.
What is the difference between volume and area?
Area covers a 2D surface (length × width, square units); volume fills 3D space (length × width × height, cubic units).
What is the difference between volume and capacity?
Volume is the space an object takes up; capacity is what a container can hold. Wall thickness makes capacity slightly smaller than exterior volume.
How do I calculate a sphere’s volume?
V=4/3·π·r³. A 5 m sphere gives 4/3·π·125≈523.60 m³, about 523,599 liters.
How do I calculate a cylinder’s volume?
V=π·r²·h. Radius 3 m and height 10 m give π·9·10≈282.74 m³.
How do I calculate a cone’s volume?
V=1/3·π·r²·h — exactly one third of its cylinder. Radius 4 m, height 9 m gives ≈150.80 m³.
How do I calculate a cube or box?
Cube: a³. Box: l·w·h. A 5 m cube holds 125 m³; a 2×3×4 m tank holds 24 m³ (24,000 L).
How do I calculate a pyramid?
Any pyramid is 1/3·base-area·height. Square pyramid: 1/3·a²·h. Base 6 m with height 9 m gives 108 m³.
How do I calculate an ellipsoid?
V=4/3·π·a·b·c from the three semi-axes. Equal axes reduce to the sphere formula.
What is a capsule volume?
Cylinder plus sphere: π·r²·h+4/3·π·r³. With r=2 m and cylinder height 5 m: ≈97.39 m³.
What is a spherical cap?
The slice of a sphere above a cut plane: V=1/3·π·h²·(3R−h). Give any two of base radius r, ball radius R, and height h and the solver derives the third.
What is a conical frustum?
A truncated cone: V=1/3·π·h·(r²+rR+R²). Top 2 m, bottom 4 m, height 5 m gives ≈146.61 m³.
How do I calculate tube (pipe) volume?
Material volume is π·(D²−d²)/4·L. A 10 cm outer, 8 cm inner, 2 m pipe holds ≈0.00565 m³ of wall material; bore capacity uses the inner diameter only.
How do I convert cubic meters to liters?
Multiply by 1,000. So 2.5 m³ = 2,500 L = 2,500,000 mL.
How many gallons are in a cubic foot?
One cubic foot ≈ 7.48052 US gallons. One US gallon ≈ 0.133681 ft³ and ≈ 3.78541 L.
Which length unit should I pick?
Any — the solver converts every dimension to meters before applying the formula. Just keep all inputs in the same selected unit.
How do I measure radius vs diameter?
Radius is half the diameter. Entering a diameter as radius overstates sphere volume 8×, so check which the tape gave you.
How do I find radius from a known volume?
Invert the formula. Sphere: r=∛(3V/4π). A 100 m³ sphere has r≈2.88 m. Cylinder: h=V/(π·r²).
How do I find volume of an irregular object?
Use displacement: submerge it and measure displaced water (Archimedes), or 3D-scan, or mass/density via V=m/ρ.
How do I size a rectangular fish tank in gallons?
Compute l·w·h in feet for ft³, then multiply by 7.48052. A 4×2×2 ft tank is 16 ft³ ≈ 119.7 gallons.
How much concrete do I need for a slab?
Slab volume is l·w·thickness in consistent units. A 10×12 ft slab 4 in thick is 40 ft³ ≈ 1.48 yd³; order 5-10% extra for waste.
How do I calculate pool volume?
Rectangular: l·w·average-depth. Circular: π·r²·average-depth. Kidney shapes: split into rectangles plus elliptical ends and sum.
What is specific volume vs density?
Specific volume (m³/kg) is the reciprocal of density (kg/m³). Both shift with temperature and pressure for fluids.
Metric vs US volume units?
Metric steps by 1000 (m³→L→mL). US uses gallons=4 qt=8 pt=16 cups=128 fl oz, plus ft³/in³/yd³ for bulk.
Why is my tube result zero?
Either a dimension is missing/zero or inner ≥ outer diameter. The wall needs D>d>0 and positive length.
How precise should inputs be?
Volume cubes linear error: a 1% length error becomes ~3% volume error. Measure diameters twice and keep consistent units.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Volume
3D space occupied, in cubic units.
Capacity
What a container can hold (interior volume).
Cubic meter
SI volume: 1 m³ = 1000 L.
Frustum
Truncated cone between two parallel planes.
Spherical cap
Slice of a sphere above a cutting plane.
Capsule
Cylinder plus two hemispheres.
Ellipsoid
Sphere scaled by axes a, b, c.
Displacement
Volume via submerged fluid rise (Archimedes).
Specific volume
Volume per mass (m³/kg), reciprocal of density.
Unit factor
Meters per selected length unit.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

NIST SI volume and length guidanceUS customary liquid-volume definitions (gal/qt/pt/cup/fl oz)

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Page Information

Last Updated:

2026-08-07

Version:

2.0.0

License:

Free to use for educational and professional purposes.