What unit does the volume result use?

Whatever linear unit you input, cubed. Enter dimensions in metres → volume in cubic metres. Inches → cubic inches. Feet → cubic feet. The calculator is unit-agnostic; it just multiplies the numbers. Use a separate unit converter if you need to switch (1 cu m = 35.31 cu ft = 1,000 litres = 264.2 US gallons).

When do I need surface area instead of volume?

Volume is for filling: concrete, mulch, water, gas. Surface area is for covering: paint, wallpaper, plating, insulation, heat-loss calculations. A swimming pool needs volume for water capacity and surface area for the liner. The calculator surfaces both for closed shapes (cube, prism, cylinder, sphere, cone, pyramid) so you don't run two separate calculations.

How do I find volume for a shape that isn't on the list?

Decompose into the eight primitives. A propane tank is typically a cylinder with two hemispherical ends — cylinder volume + 1 full sphere volume. An L-shaped pool is two rectangular prisms added. A hopper bin is a rectangular prism plus a frustum below it. The 'sum of primitives' approach handles every shape solid geometry classifies as a polyhedron or a region bounded by quadric surfaces.

What's the difference between a cone and a pyramid?

A cone has a circular base; a pyramid has a polygonal base (this calculator assumes a square base). Both use the (1/3) × base × height pattern — the cone's base is π × r², the square pyramid's base is base-edge². The factor of 1/3 reflects the same calculus identity in both shapes: a cone or pyramid is one-third the volume of the prism that fits around it.

Why is the sphere formula 4/3 × π × r³?

Derived by Archimedes (≈250 BCE) using the method of exhaustion — the geometric precursor to integral calculus. He proved that a sphere occupies exactly 2/3 of its smallest enclosing cylinder; since the cylinder's volume is 2 × π × r³, the sphere is (2/3) × 2 × π × r³ = (4/3) × π × r³. Archimedes was so proud of this proof he asked for a sphere-in-cylinder diagram on his tombstone.

Can the calculator handle very small (drops) or very large (lakes) volumes?

Yes — JavaScript numbers handle values from 5×10⁻³²⁴ to 1.8×10³⁰⁸ before precision degrades. For everyday use, enter in any consistent unit. For drinking-water dosing or pharmaceutical volumes, work in millilitres or microlitres. For lakes and reservoirs, work in cubic metres or acre-feet. The math is identical at every scale.

What is a conical frustum and when would I use one?

A frustum is the 3D shape you get when you slice the top off a cone with a plane parallel to the base — like a paper coffee filter, a lampshade, a bucket, a flowerpot, or the top of a grain silo. The formula V = (1/3) × π × h × (R² + R×r + r²) collapses back to the cone formula when r = 0 (full cone) and to the cylinder formula when R = r.

How accurate is the π approximation?

JavaScript's Math.PI is accurate to 15 decimal places (3.141592653589793) — more than sufficient for any real-world volume calculation. Results are rounded to 4 decimal places in the display. A sphere with radius 1 returns V ≈ 4.1888 cu units (the true value to 12 places is 4.188790204786 cu units — well within machine precision).

How do I convert volume to weight for ordering material?

Multiply volume by the material's density. Concrete: ~150 lb/cu ft = 4,050 lb/cu yd (a cubic yard weighs ~2 tons). Topsoil: ~75 lb/cu ft. Water: 62.4 lb/cu ft (8.34 lb/gallon). Mulch: 20–40 lb/cu ft (depends on type). For project planning, compute volume here, then look up the density and multiply. The Home Improvement calculators do this conversion for you on specific materials.

Why does the frustum require R ≥ r?

By convention, the larger radius is the bottom and the smaller is the top. If you enter R smaller than r, the calculator flags it as a likely input error rather than silently computing the same volume (frustum volume is symmetric in R and r, but the conventional drawing puts the larger radius on the bottom). Swap them mentally if your frustum is upside-down — the volume is the same.

Where do these volume formulas come from historically?

Egyptian Moscow Mathematical Papyrus (≈1850 BCE) contains the correct formula for the volume of a square frustum — predating Greek geometry by 1,500 years. Archimedes (3rd century BCE) derived the sphere formula. Cavalieri's Principle (1635) gave a general method that handles all the shapes in this calculator. Modern integral calculus (Newton, Leibniz, late 1600s) reduces every volume formula to a single integration.

What is the surface area used for?

Surface area drives material that coats or insulates the shape: paint on a cylindrical tank, polish on a sphere, drywall sheets, heat-loss calculations on a building envelope, plating thickness on metalwork. Volume drives material that fills the shape: concrete in a pour, water in a pool, gas in a propane tank, soil in a flowerpot. Real projects often need both.

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Free Volume Calculator — Cube, Cylinder, Sphere, Cone, Pyramid, Ellipsoid, Frustum

Compute the volume of eight common 3D shapes from their dimensions — and get the surface area where the shape supports it. Step-by-step working shown for every result.

Instant result
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Reviewed by CalcBold EditorialLast verified May 18, 2026Methodology

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What is Volume?

Volume is the measure of how much three-dimensional space a solid object occupies. It is always reported in cubic units — cubic metres, cubic feet, cubic inches — because volume combines three linear measurements into a measurement of capacity. Whether you are pouring concrete, ordering mulch, sizing a water tank, shipping a package, or specifying a propane bottle, every project that fills 3D space with material needs the volume first.

Volume and capacity are the same thing measured from different sides — one is the space the solid takes up, the other is the space the solid can hold. They are numerically identical, which is why a 1-cubic-metre tank holds 1,000 litres of water (1 cu m = 1,000 L) and a 1-cubic-foot box holds about 7.48 US gallons. The calculator outputs cubic-linear-units; you convert to litres, gallons, bushels, or barrels with a unit converter.

The Eight Shape Formulas

Cube & rectangular prism

V = side³                          (cube) V = length × width × height        (rectangular prism)

A cube is a rectangular prism with all three edges equal — same formula, single dimension. Both shapes have closed-form surface areas: 6 × side² for a cube; 2 × (lw + wh + lh) for a rectangular prism.

Source:NIST DLMF — Elementary Geometry· National Institute of Standards and Technology

Cylinder, sphere & cone

V = π × r² × h                     (cylinder) V = (4/3) × π × r³                 (sphere) V = (1/3) × π × r² × h             (cone)
Sphere = 2/3 of its smallest enclosing cylinder (Archimedes, ≈250 BCE)

The cylinder is the prism of revolution — area × height. The sphere is exactly 2/3 of the cylinder that just contains it. The cone is exactly 1/3 of the cylinder with the same base and height. These three shapes are linked by a single proportional relationship: 1 : 2 : 3 in volume for cone : sphere : cylinder when r and h are matched.

Source:Wolfram MathWorld — Sphere· Wolfram Research

Pyramid, ellipsoid & frustum

V = (1/3) × base² × height                  (square pyramid) V = (4/3) × π × a × b × c                    (ellipsoid) V = (1/3) × π × h × (R² + R×r + r²)          (conical frustum)

The square pyramid is the polygonal cousin of the cone — same 1/3 multiplier. The ellipsoid generalizes the sphere with three independent semi-axes; when a = b = c = r, the formula collapses back to (4/3)πr³. The frustum is a cone with the top sliced off; when r = 0, it collapses back to the cone formula, and when r = R, it becomes a cylinder.

Source:Wolfram MathWorld — Solid Geometry· Wolfram Research

Three Worked Examples

Concrete scenarios with specific numbers. Plug any of them into the calculator above to reproduce the step-by-step working.

Example 1

Pouring a rectangular concrete pad

Shape: Rectangular prism
Length: 20 ft
Width: 12 ft
Thickness: 0.333 ft (4 inches)

Apply the rectangular-prism volume formula.
V = length × width × height = 20 × 12 × 0.333
Compute.
V = 80 cu ft
Convert to cubic yards (concrete is sold by the yard).
80 / 27 ≈ 2.96 cu yd
Round up for waste — order at least 3 cu yd.
Order: 3 cu yd

Pad volume: 80 cu ft = ~2.96 cu yd. Order 3 cu yd (round up; running out mid-pour weakens the slab).

Most concrete suppliers sell in 1/4-yard increments and charge a short-load fee under 3 yd. Round up to the next quarter even if the math says otherwise.

Example 2

Sizing a cylindrical water tank

Shape: Cylinder
Radius: 2 ft (4-ft diameter)
Height: 6 ft

Apply the cylinder volume formula.
V = π × r² × h = π × 2² × 6
Compute the cross-section area first.
π × 4 ≈ 12.566 sq ft
Multiply by height.
12.566 × 6 ≈ 75.4 cu ft
Convert to US gallons (1 cu ft = 7.481 gal).
75.4 × 7.481 ≈ 564 US gallons

Tank volume: ~75 cu ft = ~564 US gallons. Surface area (for liner) = 2πr(r+h) ≈ 100.5 sq ft.

Tank ratings typically quote 'water capacity at 90%' — fill volume is usually 10% below the geometric capacity to leave headspace. Plan for 510 gal of usable water in this example.

Example 3

Mulching a conical garden mound

Shape: Cone
Radius: 5 ft
Height: 3 ft

Apply the cone volume formula.
V = (1/3) × π × r² × h = (1/3) × π × 25 × 3
Compute.
V = (1/3) × π × 75 = π × 25 ≈ 78.54 cu ft
Convert to cubic yards (mulch sold by yard or bag).
78.54 / 27 ≈ 2.91 cu yd
Convert to 2-cu-ft bags (standard retail size).
78.54 / 2 ≈ 40 bags

Mound volume: ~78.5 cu ft = ~2.9 cu yd = ~40 bags of mulch.

Bagged mulch costs ~3× as much per cubic yard as bulk delivery. For >3 cu yd, get bulk-delivered; for <1 cu yd, bagged is more convenient.

How to Use This Calculator

Pick the shape from the dropdown. The dimension fields below will automatically relabel to ask for exactly what that shape needs.
Enter each dimension in whatever linear unit you have — centimetres, inches, feet, metres. The volume output uses the same unit, cubed. Use the unit converter to switch to gallons, litres, or cubic yards if needed.
Read the volume at the top. For closed shapes (cube, prism, cylinder, sphere, cone, pyramid), the surface area is also reported below — useful for paint or insulation alongside the fill calculation.
Tap Show working to see the formula substituted with your numbers, step by step. Useful for homework and double-checking.

Volume Formulas at a Glance

The eight 3D shapes this calculator supports, paired with their volume formulas and the surface areas it also reports where applicable.

Shape reference

Volume + surface-area formulas for eight common 3D shapes

Volume + surface-area formulas for eight common 3D shapes
Scenario	Volume formula	Surface area	Typical use
Cube	V = side³	6 × side²	Storage boxes · ice · grain bins
Rectangular prism	V = l × w × h	2(lw + wh + lh)	Rooms · pools · shipping containers
Cylinder	V = π × r² × h	2π × r × (r + h)	Tanks · pipes · cans · barrels
SphereRecommended	V = (4/3) × π × r³	4π × r²	Balls · tanks · planets
Cone	V = (1/3) × π × r² × h	π × r × (r + slant)	Mulch piles · funnels · ice cream
Square pyramid	V = (1/3) × base² × h	base² + 2 × base × slant	Roofs · sculptures · containers
Ellipsoid	V = (4/3) × π × a × b × c	(no closed form)	Eggs · planets · oval tanks
Conical frustum	V = (1/3) × π × h × (R²+Rr+r²)	(complex)	Lampshades · buckets · grain hoppers

The sphere is highlighted as the volume:surface-area champion — among all closed shapes, the sphere encloses the most volume per unit of surface area (the 3D version of the isoperimetric inequality). This is why bubbles are spherical and pressure tanks are spherical or cylindrical.

Common Mistakes When Calculating Volume

Using diameter as the radius. Cylinder, sphere, cone, and frustum formulas all use radius — not diameter. A 12-inch diameter pipe has a 6-inch radius. Forgetting to halve quadruples the cylinder cross-section and the volume.
Mixing units across dimensions.Don’t put length in feet and height in inches. Convert to the same unit first. The calculator does not unit-check.
Mixing volume units with mass.“A cubic yard of concrete weighs ~2 tons” — volume × density = mass. The calculator gives volume; multiply by the material’s density to get weight for shipping or load calculations.
Forgetting the 1/3 multiplier for cones and pyramids. The naive mistake is to use the prism formula (base × height) and overestimate by 3×. Always apply the 1/3 for pointed shapes.
Using slant height instead of perpendicular height. Cone, pyramid, and frustum formulas use the perpendicular height from base to apex — not the slant length along the surface. Slant is for the surface-area formula, not the volume.
Skipping the waste factor on real-world orders. Concrete waste 10%, mulch waste 5%, sand and gravel waste 10–15%. The calculator gives you the theoretical minimum; the order should be 10–15% above for material projects.
Confusing fill volume with overall volume. Tank ratings often quote fill volume at 90% capacity. The calculator gives geometric volume — match it to fill volume by multiplying by 0.9 (or whatever fill fraction the design specifies).

Volume Terminology — Quick Reference

Eight terms that appear across solid-geometry problems and real-world fill / capacity tasks. Skim the snippet line; expand the card for the longer explanation.

Quick reference

Volume glossary

Volume Three-dimensional capacity of a solid. Always reported in cubic units (m³, ft³, in³).: Computed as the product of three linear dimensions for prisms, as a function of one dimension and constants for spheres, and as combinations for cones, pyramids, and frustums.
Surface Area The total area of every exterior face of a solid. Reported in square units (m², ft², in²).: Used for paint, wallpaper, insulation, plating, and heat-loss calculations. Sphere, cube, prism, cylinder, cone, and pyramid all have closed-form surface areas. The ellipsoid does not — there is no exact formula, only approximations like the Knud Thomsen estimate.
Capacity How much a container can hold. Numerically identical to volume for the same shape.: Capacity is the application-layer term (litres, gallons, bushels) while volume is the geometric term (m³, ft³, in³). Convert between them using a fixed multiplier: 1 m³ = 1,000 L = 264.2 US gal = 35.31 ft³.
Cubic Units The unit of volume — always the linear input unit, cubed (cu m, cu ft, cu in).: If you input centimetres, the volume comes back in cubic centimetres. Conversions: 1 cu ft = 0.0283 cu m, 1 cu yd = 27 cu ft, 1 US gal = 0.1337 cu ft. The Britannica article on volume has the full reference table.
Source: Britannica — Volume
Cavalieri's Principle Two solids with the same cross-section area at every height have the same volume. Proven by Italian mathematician Bonaventura Cavalieri in 1635.: Practical use: a leaning stack of coins has the same volume as a straight stack of identical coins. This is why a slanted cylinder has the same volume as an upright one of equal radius and perpendicular height. Cavalieri's principle is one of the foundations of integral calculus.
Frustum A cone or pyramid with the top sliced off by a plane parallel to the base. Latin for 'piece'.: Examples: paper coffee filter (cone frustum), lampshade, flowerpot, bucket, top of a grain silo. The volume formula V = (1/3) × π × h × (R² + Rr + r²) handles all conical frustums; the analogous square-pyramid frustum uses (1/3) × h × (A + B + √(AB)) where A and B are the two base areas.
Cross-Section The 2D shape revealed when a 3D solid is sliced by a plane. Drives Cavalieri's principle and integral volume calculation.: A cylinder sliced perpendicular to its axis reveals a circle (the cross-section). A cone sliced parallel to its base reveals a smaller circle. Volume can always be computed as the integral of cross-sectional area along the axis — the basis of disk and shell methods in calculus.
Density Mass per unit volume. Multiply volume by density to get the weight of a material.: Water: 1,000 kg/m³ (62.4 lb/ft³). Concrete: ~2,400 kg/m³ (150 lb/ft³). Steel: ~7,850 kg/m³. Always look up the specific density for your material before ordering — densities vary by composition, moisture, and packing.
Source: NIST — Material density references

When to Use Volume vs Surface Area

Real projects rarely need just one. Match the measurement to the task:

Volume — for material that fills 3D space: concrete, mulch, gravel, water, gas, soil, grain, oil, paint thickness × surface area.
Surface area — for material that coats the outside: paint, wallpaper, insulation, plating, heat-loss calculations on a building envelope, tank-liner sizing.
Both, multiplied together— when you need a thin layer of material over a surface, that’s actually a volume calculation: surface area × thickness. Painting a wall: area × paint thickness (~0.003 in for one coat) = volume of paint. Use the dedicated paint calculator or the concrete calculator for these compound calculations.

Background

A Brief History of Volume Measurement

Volume measurement predates written mathematics. The Moscow Mathematical Papyrus (≈1850 BCE) contains the correct formula for the volume of a square pyramidal frustum — V = (1/3) × h × (a² + a×b + b²) — predating Greek geometry by 1,500 years and demonstrating that the Egyptians understood the (1/3) multiplier for tapered solids without modern calculus ^[1].

Archimedes (3rd century BCE) achieved the most famous volume proof in mathematics: a sphere occupies exactly 2/3 of its smallest enclosing cylinder. He arrived at this using the method of exhaustion — inscribing and circumscribing polyhedra and bounding the sphere's volume between them. He considered this his greatest result and asked that a sphere-and-cylinder diagram be carved on his tomb ^[2]. The proof also handed mathematics the canonical (4/3)πr³ formula for the sphere.

Modern volume measurement standards come from the metric system established under Napoleon (1799) and codified internationally by the Bureau International des Poids et Mesures (BIPM) ^[3]. The cubic metre is the SI derived unit; the litre is exactly 1/1000 of a cubic metre; the kilogram of water at 4°C occupies exactly one litre by historical definition. In the United States, the gallon definition was standardised in 1893 alongside the survey foot — the US gallon (3.785 L) is smaller than the imperial gallon (4.546 L), causing recurring confusion in international supply chains.

MacTutor — Egyptian Mathematics + Moscow Mathematical Papyrus · MacTutor History of Mathematics, University of St Andrews
Archimedes — On the Sphere and Cylinder · MacTutor History of Mathematics, University of St Andrews · 3rd c. BCE
BIPM SI Brochure — The International System of Units · Bureau International des Poids et Mesures

Related Math Tools

Volume is the 3D companion to area. Pair this calculator with the area calculator when you need the flat-surface view, the unit converter when switching between cubic feet and gallons or cubic metres and litres, and the percentage calculator when adding waste factors or fill-fraction adjustments. For material-volume tasks downstream of the geometric volume (concrete, paint, mulch), use the dedicated Home Improvement calculators — concrete, mulch (coming soon), paint, drywall. They bake the density / coverage / waste factor into one step.

Sources & Methodology

The formulas, thresholds, and benchmarks behind this calculator are anchored to the primary sources below. Where a study or agency document is the underlying authority, we link straight to it — not a summary or republished version.

NIST DLMF — Elementary Geometry· National Institute of Standards and Technology
Authoritative reference for the elementary functions (π, cubes, square roots) that underlie every volume and surface-area formula in the calculator.
Accessed 2026-05-18
Wolfram MathWorld — Solid Geometry· Wolfram Research
Standard reference for the algebraic identities producing volume and surface area of cubes, rectangular prisms, cylinders, spheres, cones, pyramids, ellipsoids, and frustums.
Accessed 2026-05-18
BIPM SI Brochure — Units of Volume· Bureau International des Poids et Mesures
Defines cubic metre as the SI derived unit of volume and the conventions for compound units (litre, cubic foot, cubic yard) used in the calculator's unit-agnostic output.
Accessed 2026-05-18
Britannica — Geometry: Volumes of Solids· Encyclopaedia Britannica
Historical overview of volume formulas — Archimedes' On the Sphere and Cylinder (sphere = 2/3 of circumscribing cylinder), Cavalieri's principle, integration as the modern foundation.
Accessed 2026-05-18

Frequently Asked Questions

The most common questions we get about this calculator — each answer is kept under 60 words so you can scan.

What unit does the volume result use?
Whatever linear unit you input, cubed. Enter dimensions in metres → volume in cubic metres. Inches → cubic inches. Feet → cubic feet. The calculator is unit-agnostic; it just multiplies the numbers. Use a separate unit converter if you need to switch (1 cu m = 35.31 cu ft = 1,000 litres = 264.2 US gallons).
When do I need surface area instead of volume?
Volume is for filling: concrete, mulch, water, gas. Surface area is for covering: paint, wallpaper, plating, insulation, heat-loss calculations. A swimming pool needs volume for water capacity and surface area for the liner. The calculator surfaces both for closed shapes (cube, prism, cylinder, sphere, cone, pyramid) so you don't run two separate calculations.
How do I find volume for a shape that isn't on the list?
Decompose into the eight primitives. A propane tank is typically a cylinder with two hemispherical ends — cylinder volume + 1 full sphere volume. An L-shaped pool is two rectangular prisms added. A hopper bin is a rectangular prism plus a frustum below it. The 'sum of primitives' approach handles every shape solid geometry classifies as a polyhedron or a region bounded by quadric surfaces.
What's the difference between a cone and a pyramid?
A cone has a circular base; a pyramid has a polygonal base (this calculator assumes a square base). Both use the (1/3) × base × height pattern — the cone's base is π × r², the square pyramid's base is base-edge². The factor of 1/3 reflects the same calculus identity in both shapes: a cone or pyramid is one-third the volume of the prism that fits around it.
Why is the sphere formula 4/3 × π × r³?
Derived by Archimedes (≈250 BCE) using the method of exhaustion — the geometric precursor to integral calculus. He proved that a sphere occupies exactly 2/3 of its smallest enclosing cylinder; since the cylinder's volume is 2 × π × r³, the sphere is (2/3) × 2 × π × r³ = (4/3) × π × r³. Archimedes was so proud of this proof he asked for a sphere-in-cylinder diagram on his tombstone.
Can the calculator handle very small (drops) or very large (lakes) volumes?
Yes — JavaScript numbers handle values from 5×10⁻³²⁴ to 1.8×10³⁰⁸ before precision degrades. For everyday use, enter in any consistent unit. For drinking-water dosing or pharmaceutical volumes, work in millilitres or microlitres. For lakes and reservoirs, work in cubic metres or acre-feet. The math is identical at every scale.
What is a conical frustum and when would I use one?
A frustum is the 3D shape you get when you slice the top off a cone with a plane parallel to the base — like a paper coffee filter, a lampshade, a bucket, a flowerpot, or the top of a grain silo. The formula V = (1/3) × π × h × (R² + R×r + r²) collapses back to the cone formula when r = 0 (full cone) and to the cylinder formula when R = r.
How accurate is the π approximation?
JavaScript's Math.PI is accurate to 15 decimal places (3.141592653589793) — more than sufficient for any real-world volume calculation. Results are rounded to 4 decimal places in the display. A sphere with radius 1 returns V ≈ 4.1888 cu units (the true value to 12 places is 4.188790204786 cu units — well within machine precision).
How do I convert volume to weight for ordering material?
Multiply volume by the material's density. Concrete: ~150 lb/cu ft = 4,050 lb/cu yd (a cubic yard weighs ~2 tons). Topsoil: ~75 lb/cu ft. Water: 62.4 lb/cu ft (8.34 lb/gallon). Mulch: 20–40 lb/cu ft (depends on type). For project planning, compute volume here, then look up the density and multiply. The Home Improvement calculators do this conversion for you on specific materials.
Why does the frustum require R ≥ r?
By convention, the larger radius is the bottom and the smaller is the top. If you enter R smaller than r, the calculator flags it as a likely input error rather than silently computing the same volume (frustum volume is symmetric in R and r, but the conventional drawing puts the larger radius on the bottom). Swap them mentally if your frustum is upside-down — the volume is the same.
Where do these volume formulas come from historically?
Egyptian Moscow Mathematical Papyrus (≈1850 BCE) contains the correct formula for the volume of a square frustum — predating Greek geometry by 1,500 years. Archimedes (3rd century BCE) derived the sphere formula. Cavalieri's Principle (1635) gave a general method that handles all the shapes in this calculator. Modern integral calculus (Newton, Leibniz, late 1600s) reduces every volume formula to a single integration.
What is the surface area used for?
Surface area drives material that coats or insulates the shape: paint on a cylindrical tank, polish on a sphere, drywall sheets, heat-loss calculations on a building envelope, plating thickness on metalwork. Volume drives material that fills the shape: concrete in a pour, water in a pool, gas in a propane tank, soil in a flowerpot. Real projects often need both.