Volume Calculator — 24 Shapes with Interactive 360° 3D Views, Weight, Buoyancy & Comparisons

What volume actually measures

Volume is the amount of three-dimensional space a solid occupies. Where length measures a line, and area measures a flat surface, volume measures the inside of a solid — how much water a container holds, how much material is needed to build it, or how much mass it will have at a given density. Volume is always measured in cubic units: cubic centimeters (cm³), cubic meters (m³), cubic inches (in³), cubic feet (ft³) — or in liquid measures like liters, milliliters, gallons, cups, and fluid ounces.

Every volume formula traces back to one idea: base area × height, or its calculus generalization, the integral of cross-sectional area along the height. A rectangular box is a rectangle (base) extruded straight up. A cylinder is a circle extruded straight up. A pyramid and a cone are a prism or cylinder where the top has tapered to a point — which is why their volume is exactly one-third of the shape they came from. A sphere is the limit of stacked circular disks and produces the (4/3)πr³ term that shows up everywhere in science and engineering.

Because every shape in this calculator reduces to that same idea — integrate area along an axis — you can treat volume as one unified concept. The formulas differ because the shapes differ; the underlying mathematics is identical. This calculator gives you both the plain-language result and the exact formula behind it, so you can either trust the number or follow the derivation.

How to use this calculator

1. Pick a shape. Use the "Most used shapes" row for the six most common shapes (cube, sphere, cylinder, box, cone, hemisphere). Or filter by category — Spheres, Cylinders, Prisms, Pyramids, Platonic — to narrow the full grid. Clicking any shape immediately updates the diagram and inputs.
2. Orbit in 3D. Drag the 3D view in any direction to rotate the shape in 360°. Scroll to zoom between 40% and 400%; on touch devices, use the zoom buttons. Click ⟳ to toggle auto-rotate for a slow spin, ↺ to reset the view, and □ for a wireframe overlay that shows the mesh structure.
3. Enter dimensions. Every shape has its own input fields — radius for a sphere, radius and height for a cylinder, length × width × height for a box. The 3D view scales live, so you see the actual proportions of what you're measuring.
4. Use a quick pick. Below the fields are three preset dimension buttons for each shape — for a sphere, r=1, r=5, r=10 cm. Click any to load instantly. These are the values most people actually need (a marble, an orange, a soccer ball).
5. Reverse mode. Set the Solve For dropdown to any dimension instead of "Volume" — the volume becomes an input and the missing dimension is computed. Useful for "how tall is a 500 mL can?" or "what radius makes a 5 L sphere?"
6. Choose your units. The Input unit applies to every dimension (mm, cm, m, in, ft, yd). The Primary output unit controls the big result card, and five other common units appear alongside — so you get liters, mL, gallons, fluid ounces, and cups all at once without doing any conversion yourself.
7. Read the volume and surface area. The result banner shows your volume in your chosen unit, five secondary unit cards, and the surface area of the shape (which is a different quantity — square units, not cubic).
8. Material weight. Open the Material Weight panel, pick from 60+ materials grouped by category (water, metals, building, wood, plastics, gases, special), and see the mass in kg, g, lb, oz, and metric tons — all updated live.
9. Buoyancy. The buoyancy panel compares your shape's mass against the mass of water it displaces. It shows a density bar comparison, floats-or-sinks verdict, net force, and how far under water a floating object would sit.
10. Cost. Enter a price per kilogram and see the total cost of the shape in that material. Includes a full breakdown of volume, density, mass, and price.
11. Fill time. Enter a flow rate and see how long the shape would take to fill (or drain) at that rate, in seconds, minutes, hours, or days.
12. Batch mode. Add multiple shapes to a batch list, get the total volume across all of them, and download as CSV for later use.
13. History. Your last 50 calculations are saved to this browser's localStorage and grouped by Today / Yesterday / Older. Click any entry to reload it exactly.

Volume and surface area formulas for all 24 shapes

Every shape supported by this calculator, with the exact volume and surface area formula, and a worked example showing the number you'd get for the default dimensions.

Why cone and pyramid are exactly one-third

This is one of the most surprising results in elementary geometry, and it holds for any pyramid or cone regardless of its base shape. Fill a cone with water and pour it into the cylinder of the same base and height — you can pour three times before it fills. The same is true of a pyramid and the prism it fits inside.

The reason is calculus, but you can reason it out intuitively: at each height up the cone, the cross-section shrinks linearly. At the top, the radius is 0. Halfway up, the radius is half. That shrink factor is (1 − y/h), and squaring it (because area goes as radius squared) gives (1 − y/h)². Integrate that from 0 to h and you get h/3 — the one-third that shows up in the formula.

Historically, this result was proven without calculus by Democritus (around 400 BCE) and made rigorous by Eudoxus of Cnidus using the method of exhaustion. Cavalieri's principle in 1635 made the proof much simpler using infinite slices, and Newton and Leibniz placed it on the foundation of integration in the 1660s.

Volume vs surface area — two very different things

Volume measures what is inside the shape. Surface area measures what you would need to cover the outside. A cubic meter of water weighs about 1,000 kilograms — but the surface area of a 1 m cube is 6 m², more than a small table. Every shape in this calculator displays both, and confusing them is one of the most common mistakes in geometry homework.

These two properties behave in opposite ways as you change size. If you double the linear dimensions of a shape, its surface area grows by 2² = 4× but its volume grows by 2³ = 8×. This "square-cube law" is why:

• Small animals can jump proportionally much higher than large animals — their mass (∝ volume) grows faster than their muscle cross-section (∝ area).
• Hot objects cool faster when they are small — their surface-to-volume ratio is higher.
• Giant insects would suffocate — their lungs cannot get enough oxygen through their surface area at large volumes.
• A scaled-up model boat does not float the way you would expect — its weight grows three times faster than its water displacement surface.
• Spacecraft re-entering the atmosphere need heat shields — the mass-to-surface ratio determines heating rate.
• Medicine dosages for children use body weight (∝ volume), not surface area — because drug distribution depends on tissue mass.

This calculator shows both volume and surface area for every shape, so you can see the relationship directly. Change the dimensions and watch the two values grow at different rates.

Volume vs capacity vs mass — three quantities that get confused

Everyday language treats volume and capacity as the same thing, but they measure different things. Volume is the space a solid occupies. Capacity is how much a container can hold. For a solid object — a brick, a golf ball, a block of wood — the two are identical: the object's capacity to hold things is zero, but its volume is the space it fills.

For a container — a bottle, a tank, a swimming pool — capacity is the internal volume. It's always smaller than the external volume because of wall thickness. A "1 L bottle" typically has an external volume of about 1.05 to 1.15 L depending on the bottle's material and wall thickness. This is also why a "12 oz" soda can holds 355 mL of liquid but physically occupies about 400 mL of space.

Mass is entirely different: it measures the amount of matter, not the amount of space. Mass depends on both volume and density. A cubic meter of water has a mass of about 1,000 kg. A cubic meter of air is 1.2 kg. A cubic meter of gold is 19,300 kg. Same volume, wildly different mass. The Material Weight panel in this calculator handles that conversion for 60+ materials — mass = volume × density.

Volume units and how they convert

Volume is the least-standardized measurement family in everyday use. Metric countries quote liters for liquids but inches and feet for construction, gallons for fuel, cups for cooking, and cubic centimeters for engine displacement. Here are the conversions this calculator uses, all exact or NIST-verified:

⚠ UK vs US gallons. A US gallon is 3.78541 liters. A UK (imperial) gallon is 4.54609 liters — about 20% larger. When a recipe or vehicle spec says "gallon" without a country, it usually means the local convention (US in the Americas and most of Asia, UK in the UK, Ireland, and some Commonwealth countries). Mixing the two is one of the most common sources of error in cooking and fuel calculations.

Understanding volume units deeply

Why does a liter exist when we already have cubic centimeters? Because the numbers get unwieldy fast. A cubic centimeter is one thousandth of a liter. A cubic meter is one thousand liters. Both are useful when you need cubic units for engineering, and liters when you need to compare liquid quantities in daily life.

The liter's history explains its role. It was defined in 1795 by the French revolutionary government as exactly 1 cubic decimeter (10 cm × 10 cm × 10 cm = 1,000 cm³). This was part of the metric system's attempt to make measurement rational — one liter of water weighs one kilogram, and 1,000 liters of water equals one cubic meter which weighs one metric ton. That chain of round numbers doesn't exist in the imperial system, which is why the metric system spread so widely in science.

The gallon has a much more complicated history. The US gallon is defined as 231 cubic inches — a measurement with no clean metric equivalent. The UK gallon is defined as the volume of 10 pounds of distilled water at 62 °F — a measurement tied to the weight of water, not to a geometric definition. When the British Empire switched to metric in the 1960s, the UK gallon became a legacy unit still used informally for beer and milk.

Fluid ounces and cups are similarly messy. A US fluid ounce is 1/128 of a US gallon, but a UK fluid ounce is 1/160 of a UK gallon. A US cup is 8 US fluid ounces, but a metric cup (used in Australia, Canada, and much of the Commonwealth) is 250 mL — not 236.6. A Japanese cup is 200 mL. When a recipe crosses borders, the "cup" is the most common source of error.

Worked examples from everyday life

How much water in an Olympic swimming pool?

An Olympic pool is 50 m × 25 m × 2 m (minimum depth). V = 2,500 m³ = 2,500,000 liters = 660,430 US gallons. That is about 2.5 million one-liter bottles, or the daily water use of roughly 25 average households. The pool weighs 2,500 metric tons when full, which is why Olympic pools are engineered as part of the building's structure, not just dropped onto a floor.

The volume of a soccer ball

A size-5 regulation soccer ball has a circumference of 68–70 cm, giving a radius of about 10.9–11.1 cm. Using r = 11 cm: V = (4/3) × π × 11³ ≈ 5,575 cm³ = 5.6 liters. When you kick a soccer ball you are moving about 5.6 kg of air plus 430 g of ball. A ball with a 5.6 L volume displaces 5.6 kg of water — that's why a soccer ball barely floats if you put it in a pool, with most of its volume submerged.

The volume of a soda can

A US 12-ounce soda can lists 355 mL on the label, but the can itself is not a perfect cylinder. A model using r = 3.3 cm and h = 12.2 cm gives V ≈ 417 cm³, which is more than the labeled 355 mL because the visible can holds headspace and has domed ends. The aluminum sheet of the can adds only about 15 mL of material volume — the rest is the liquid and the trapped gas.

The volume of a basketball

A men's size-7 basketball has a circumference of 75 cm, giving r ≈ 11.9 cm. V = (4/3) × π × 11.9³ ≈ 7,085 cm³ = 7.1 liters — roughly 20 US cups, or 4.5× the volume of a soccer ball. A regulation basketball is heavier than a soccer ball but larger, so its average density is lower — meaning it floats with more of its volume above water.

The volume of a golf ball

Regulation golf balls have a minimum diameter of 42.67 mm, so r = 21.34 mm. V = (4/3) × π × 21.34³ ≈ 40,684 mm³ = 40.7 cm³ — about 1.5 US fluid ounces. Despite its small size, a golf ball is denser than water (density about 1.13 g/cm³), so it sinks in fresh water. Salt water is denser, and a golf ball almost floats in the Dead Sea.

The volume of a tennis ball

A regulation tennis ball has a diameter of 65.4–68.6 mm, so r ≈ 33.5 mm. V ≈ 157 cm³, or about 5.3 US fluid ounces. That makes a tennis ball about 3.9× the volume of a golf ball and about 0.36× the volume of a baseball.

How much does a bathtub hold?

A standard bathtub is roughly 1.5 m × 0.7 m × 0.45 m of interior volume, giving V = 0.47 m³ = 470 liters. That is about 124 US gallons. In practice, the tub is not filled all the way, so a typical bath uses 150–250 liters. A deep soaking tub (Japanese ofuro style) might hold 300–400 liters.

How much water in an average human?

The average adult weighs 62 kg and has a density close to water (0.985 g/cm³ including air in the lungs). Volume ≈ 62,900 cm³ = 62.9 liters. About 63 one-liter bottles — which is why a bathtub barely fits two people, and why a swimming pool has to be much larger than it looks to be "deep enough to dive into."

How much gold in a wedding band?

A typical wedding band is a torus with major radius R = 10 mm and minor radius r = 1 mm. V = 2π² × 10 × 1² ≈ 197 mm³ = 0.197 cm³. At 19.3 g/cm³ for gold, that weighs 3.8 g — about 0.12 troy ounces. Gold's density is what makes even small rings feel heavy in the hand.

Volumes in astronomy

For fun: Earth's volume is 1.083 × 10¹² km³ (using the oblate spheroid formula with a = 6,378 km, c = 6,357 km). The Sun's volume is 1.412 × 10¹⁸ km³ — 1.3 million Earths could fit inside. Jupiter's volume is 1,321 Earths. The volume of the entire observable universe is roughly 4 × 10⁸⁰ m³, which means the "volume" of everything we can see is about 10⁸⁰ liters of space.

Common mistakes when calculating volume

• Using diameter when the formula needs radius. Every sphere, cylinder, cone, hemisphere, capsule, and torus formula uses r. If you have the diameter (a common spec on balls and pipes), halve it before entering it. This is the single most common error in volume calculations.
• Mixing units within one calculation. If you enter length in inches and height in centimeters, the result is meaningless. Convert everything to one unit first — or use this calculator's unit dropdown so all dimensions use the same base unit automatically.
• Confusing volume and surface area. 100 cm³ of water is not the same as 100 cm² of paint coverage. Volume is inside, surface area is outside, and they have different units (cubic vs square).
• Using the wrong "height" for a cone. The height h in V = (1/3)πr²h is the perpendicular height (from base to tip), not the slant height. Slant height is longer and produces an inflated volume — sometimes by 20–40%.
• Forgetting the third dimension in volume units. Writing "100 cm" for a volume makes no sense. Volume is cubic: cm³, m³, in³, ft³. Length is linear, area is square, volume is cubic — the exponent matches the dimension count.
• Assuming all "ounces" are the same. A fluid ounce (fl oz) measures volume — 29.57 mL in the US. A solid ounce (oz) measures mass — 28.35 g. If a recipe says "8 oz of milk" it means fluid ounces; if it says "8 oz of cheese" it means weight ounces. The two are entirely different quantities.
• Trusting the labeled capacity of a container. A "12 oz" can has nominal capacity of 12 US fl oz = 355 mL, but the physical can volume is closer to 400 mL because of headspace and dome structure. A "1 L" bottle is usually 1.05 L external.
• Using external instead of internal measurements. The internal capacity of a container is less than its external volume because of wall thickness. A pipe's water capacity uses the internal diameter; its material volume uses both internal and external diameters.

Where volume calculations come up in real life

• Shipping and freight. Couriers charge dimensional weight — the greater of actual weight and volume-based weight. Knowing cubic inches and dividing by the carrier's divisor (usually 139 for inches-to-pounds, 5000 for cm-to-kg) tells you the billed weight. This is why shipping an empty box can cost almost as much as shipping the same box full.
• Cooking and baking. US recipes use cups (236.6 mL), tablespoons (14.8 mL), and teaspoons (4.93 mL). Metric recipes use milliliters. Converting between them requires the exact volume ratios shown above, because a "cup" of flour scooped vs sifted can vary by 20% — so volumetric recipes are inherently less precise than weight-based ones.
• Aquariums and pools. Knowing the volume in liters tells you how much water treatment, salt, chlorine, or pH buffer to add — all chemicals are dosed per liter. A 200 L aquarium needs 1 mL of conditioner per 4 liters — so 50 mL total. Get the volume wrong and you overdose the fish.
• Fuel and economy. Fuel tanks are specced in liters or gallons. A cylinder-shaped tank of radius 40 cm and length 120 cm holds about 603 liters — roughly 159 US gallons. That's about 2 barrels of oil — which is why fuel tank capacities are often quoted in barrels for industrial applications.
• Construction and concrete. A concrete slab of 5 m × 4 m × 0.1 m has a volume of 2 m³, and concrete is ordered by the cubic meter. Density 2,400 kg/m³ means that slab weighs 4,800 kg — nearly 5 metric tons, which is why concrete delivery requires pouring before it cures.
• 3D printing. Filament usage is estimated from volume × infill percentage × 1.24 g/cm³ (PLA density). A 30 cm³ model at 20% infill uses about 7.4 g of filament. A full spool is usually 1 kg, giving a theoretical capacity of about 800 cm³ of printed material at 100% infill.
• Medical dosing. Liquid medications are prescribed in mL, and pediatric dosing often uses body weight (∝ volume). Mistakes in volume calculation cause real dosing errors — which is why oral syringes are marked in mL and never in teaspoons for pediatric medications.
• Astronomy and physics. The volume of the Sun, planets, and stars is directly computed from V = (4/3)πr³. Earth's volume is 1.083 × 10²¹ m³ — a number so large it's easier to compare to other planets than to imagine.
• Wine and spirits. Bottles are standardized at 750 mL (a "fifth" from the old fifth-of-a-gallon measure). A "magnum" is 1.5 L, a "jeroboam" 3 L, a "rehoboam" 4.5 L. Knowing the volume helps when planning for a party — a magnum pours 10 glasses, a standard bottle 5.
• Pharmaceutical manufacturing. Vials, ampoules, and syringes are all volume-measured. Exact fill volumes matter to the milligram — a 10% overfill of a $50 dose wastes $5 per vial across millions of units.

Volume in cooking — where units matter most

Cooking is where volume errors hurt most directly. The US cup, the metric cup, and the Japanese cup all differ:

So an American recipe asking for 1 cup (236 mL) becomes 0.94 cups if you use a metric cup (250 mL) — a 6% error, which can matter for baking but rarely for cooking. The difference between a US cup and a Japanese cup is 18%, which absolutely matters for Japanese recipes in an American kitchen. Professional bakers sidestep the problem entirely by using grams instead of cups.

Another common cooking error: measuring liquids and dry ingredients the same way. One cup of water is 236.6 mL, but one cup of flour scooped from a bin is about 150 g, while one cup of flour sifted is about 120 g — a 25% difference in mass for the same volume. Volume measurement of dry ingredients is inherently imprecise.

Volume in construction — bulk materials

Construction materials are almost always ordered by volume, and the numbers matter. Common materials and their weights per unit volume:

These densities mean that a "small dump truck load" of concrete is much heavier than the same volume of mulch. A landscaping project that needs 10 m³ of mulch requires a truck that can only carry 4 tons — but the same 10 m³ of concrete would need a 24-ton truck. Volume-based ordering requires density-based logistics.

Volume in astronomy — from planets to the observable universe

Astronomy uses the same volume formulas, just with much larger numbers. Here is a reference table of volume for objects across the astronomical scale:

The volume of the Sun is 1.3 million times Earth's. Betelgeuse, a red supergiant, is so large that if placed at the Sun's position it would engulf Mars' orbit — its volume is around 900 trillion Earths, or 1.6 × 10²⁹ km³. And the observable universe has a volume of about 4 × 10⁸⁰ m³, which is 10⁶⁰ times larger than the Sun.

A short history of volume measurement

Volume formulas have one of the longest histories in mathematics. The ancient Egyptians knew how to compute the volume of pyramids and frustums — the Moscow Mathematical Papyrus (c. 1850 BCE) gives the correct frustum formula. The Babylonians computed volumes of cylinders and prisms, and their approximations of π were accurate to about 1 part in 1,000.

The Greek mathematicians turned these practical rules into a deductive system. Eudoxus of Cnidus (c. 370 BCE) developed the method of exhaustion — an early form of integration — to prove the volume of a cone, pyramid, and sphere. Archimedes (c. 250 BCE) famously discovered that a sphere is two-thirds of the cylinder that contains it (in his work On the Sphere and Cylinder), and proved the volume of a segment of a paraboloid in The Method of Mechanical Theorems.

Archimedes considered the sphere-cylinder result his greatest achievement — he asked for it to be engraved on his tombstone. His discovery of it led to the formula V = (4/3)πr³, which remains unchanged after 2,000 years. It was the first proof that the volume of a curved solid could be exactly determined, not just approximated.

The 1635 work of Bonaventura Cavalieri — Geometria indivisibilibus continuorum nova quadam ratione promota — established that solids with equal cross-sections at equal heights have equal volumes. This principle let Newton and Leibniz unify all previous volume work into calculus in the 1660s, and every volume formula in this calculator follows from that framework.

The metric system, established during the French Revolution (1795), finally standardized liquid volume measurement across nations. The liter was defined as one cubic decimeter — a clean, geometric definition. Before the metric system, every country, region, and even town could have its own "gallon" or "pint," making international trade difficult and recipes non-transferable.

Dimensional analysis — why volume is cubic

Volume is fundamentally about the number of dimensions. A line has 1 dimension, so its "size" is measured in linear units (m, cm, ft). A surface has 2 dimensions, so its size is measured in square units (m², cm²). A solid has 3 dimensions, so its size is measured in cubic units (m³, cm³).

This pattern is universal. In 4-dimensional space, "hypervolume" would be measured in 4th-power units. In 1-dimensional space, "length" is just length. The exponent on the unit always matches the number of dimensions the shape extends into. That is why every volume formula produces a cubic result — the formula is inherently a 3-dimensional integral.

Practical consequence: a shape's volume scales as the cube of any linear scale factor. If you scale a statue 3× taller, the volume goes up 27×. If you print it 3× larger on each axis, it weighs 27× more — which is why 3D-printed sculptures have to hollow out inside, and why physical models of buildings and ships use much lighter materials than the originals.

Quick reference — real-world object volumes

Privacy

Every calculation, 3D view, and conversion runs in your browser using vanilla JavaScript. This calculator does not send your calculation inputs to our server. There is no account, no login, and no network request after the page loads. Your last 50 calculations are stored in your browser's localStorage only, and can be cleared at any time from the History panel.

Limitations

• This calculator handles solid geometric shapes with known formulas. It does not integrate over arbitrary voxel meshes, 3D scan files, or irregular solids — those require CAD or voxel-based tools.
• Surface area for the ellipsoid and both spheroids uses numerical approximations (Knud Thomsen). The error is under 1.1% for most axis ratios, but a truly flat ellipsoid may deviate by a few percent.
• Material weight uses standard room-temperature densities. Real materials vary — the density of pure water is approximately 1 g/cm³ near 4 °C, dropping to 0.958 g/cm³ at 100 °C.
• Buoyancy assumes pure water at 20 °C (998 kg/m³). Salt water is denser (1,025 kg/m³) and objects float more easily in it; the panel does not account for this.
• Reference-object volumes in the size comparison are rounded published figures; a regulation soccer ball has a specific size range and a "bathtub" varies widely by model.
• The 3D view is a wireframe render with painter's algorithm — it's not a physically-accurate ray tracer, but it shows every shape correctly in 360°.
• Reverse mode for the capsule, spheroid, and Platonic solids solves only one dimension at a time — you must know all other dimensions.
• The 3D renderer uses canvas 2D with painter's algorithm, not WebGL. Some very complex meshes (like the dodecahedron with its approximate face list) may render slightly imperfectly compared to a true 3D engine.

Frequently asked questions

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