How It Works
Pick your shape from the dropdown, then enter the measurements you have. You can use any unit—millimeters, centimeters, meters, kilometers, inches, feet, yards, or miles. The calculator converts between them automatically.
For cylinders, spheres, and cones, enter either radius or diameter—whichever measurement you know. The math works with both. Each input field has hints explaining what to measure.
After calculating, you'll see the volume, surface area, conversions to liquid measurements (liters, gallons), the formula used, and step-by-step math showing how the answer was found.
What Is Volume?
Volume is the amount of 3D space something takes up. Think of it as asking "how much can this hold?" or "how much room does this take up?" It's always measured in cubic units—cm³, m³, in³, ft³. A 5 cm cube has 125 cubic centimeters of space inside. That's 5 × 5 × 5, the total room you could fill with water or air or sand.
Volume vs. Area—What's the Difference?
Don't mix these up. Area is flat (2D)—it covers a surface, measured in square units like cm² or ft². Volume is the whole three-dimensional object, measured in cubic units. A square drawn on paper? That's area. A box sitting on your desk? That's volume. Area tells you how much paint covers something; volume tells you how much stuff fits inside it.
Volume vs. Surface Area
Here's where it gets interesting. Volume is the space inside an object—surface area is all the outside skin added together. A swimming pool's volume tells you how many gallons of water you need to fill it. Its surface area tells you how much pool paint or liner you need to cover it. Big cube? Lots of volume, relatively little surface area. Crumpled aluminum foil? Tiny volume, lots of surface area. Same object, two totally different measurements.
Cubic Units
Volume uses cubed units because you're multiplying three dimensions. 5 cm × 5 cm × 5 cm = 125 cm³, not 125 cm. Here's the trap: 1 meter = 100 centimeters, right? But 1 m³ doesn't equal 100 cm³. It equals 1,000,000 cm³ (that's 100 × 100 × 100). When converting volume, you cube the factor. Miss this and your calculations will be wildly wrong.
Liquid Volume
Liters, gallons, milliliters—these are just cubic measurements with different names. 1 liter = 1,000 cm³ exactly. 1 US gallon ≈ 3,785 cm³. Your milk jug says "1 liter" but it's really 1,000 cubic centimeters of space. A 50-gallon fish tank? That's roughly 189,000 cm³. Cubic units and liquid volumes are the same thing; you just need the conversion factors.
Why Calculate Volume?
You use volume constantly without thinking about it. Shipping costs? Based on package volume. Pouring a concrete driveway? You're ordering cubic yards. Baking a cake? Recipe volumes in cups. Pool chemicals? Depends on gallons. Medicine doses? Milliliters. Even your phone's storage is measured in volume-adjacent units. It's one of those everyday math things that shows up everywhere once you start looking.
Volume Formulas for 3D Shapes
The Simple Ones
Cube
V = s³. Take the side length and cube it (multiply it by itself three times). All edges are the same, which is why it's so simple—no juggling different measurements. A 5 cm cube? That's 5³ = 125 cm³.
What About Rectangular Prisms?
This one's V = l × w × h—length times width times height. Most common shape for boxes, rooms, containers. A 10 × 5 × 3 cm box gives you 150 cm³. Just multiply all three dimensions straight across.
Cylinders, Cans, and Circular Things
V = πr²h is really just the circle's area (πr²) times the height. Picture a stack of circles—that's your cylinder. With r = 5 cm and h = 12 cm, you get π × 25 × 12 ≈ 942 cm³. Cans, pipes, tanks all use this.
The Sphere Formula
The classic ball formula: V = (4/3)πr³. That 4/3 comes from calculus, but you just need to remember it. Radius of 5 cm? Volume is (4/3) × π × 125 ≈ 524 cm³. Planets, baseballs, bubbles—anything perfectly round.
Pyramids and Cones (The One-Third Rule)
Why divide by 3 for pyramids and cones? Here's the thing: you could fit three pyramids inside a box with the same base and height. Same deal with cones—three cones equal one cylinder. That's not an approximation, it's exact. The pointed top means you're using one-third the space of the full rectangular or cylindrical shape.
Cone
V = (1/3)πr²h—it's a cylinder divided by 3. With r = 5 cm and h = 12 cm, that's (1/3) × π × 25 × 12 ≈ 314 cm³. Ice cream cones, funnels, traffic cones.
Square Pyramid
Take the base area (s²), multiply by height, divide by 3: V = (1/3)s²h. Base of 6 cm and height of 8 cm gives (1/3) × 36 × 8 = 96 cm³. Think Egyptian pyramids or tent roofs.
Rectangular Pyramid
V = (1/3)lwh follows the same pattern. An 8 × 6 cm base with 10 cm height? (1/3) × 48 × 10 = 160 cm³.
Prisms (Constant Cross-Section)
Any prism—triangular, pentagonal, whatever—follows the same logic: base area times length.
Triangular Prism
V = (1/2)bh × L is just the triangle's area times the prism's length. Triangle base 6 cm, height 4 cm, length 10 cm? That's (1/2) × 6 × 4 × 10 = 120 cm³. Toblerone bars are the perfect example.
Weird and Wonderful Shapes
Ellipsoids—Stretched Spheres
Stretched sphere. V = (4/3)πabc where a, b, c are the three semi-axes (half-dimensions). A football with axes 5, 4, and 3 cm gives (4/3) × π × 5 × 4 × 3 ≈ 251 cm³. Earth is actually a slightly squashed ellipsoid.
Hemisphere
Half a sphere, so half the formula: V = (2/3)πr³. Radius 6 cm? Volume ≈ 452 cm³. Bowls, domes, igloos.
The Torus (Donut Shape)
V = 2π²Rr² where R is the big radius (center to tube center) and r is the tube thickness. R = 8 cm, r = 2 cm gives roughly 632 cm³. Yes, actual donuts, but also O-rings and tires.
Capsules and Pill Shapes
Pill-shaped: cylinder with hemispheres on the ends. V = πr²h + (4/3)πr³—that's the cylinder part plus a full sphere. With r = 3 cm and h = 10 cm (just the cylinder part), you get about 396 cm³ total.
Frustums (Chopped-Off Shapes)
Cone Frustum
Cone with the tip sliced off—think buckets or lampshades. V = (1/3)πh(r₁² + r₁r₂ + r₂²) accounts for both the top and bottom radii. Top r = 3 cm, bottom r = 6 cm, height = 8 cm? That's roughly 528 cm³.
Pyramid Frustum
Same idea for pyramids: V = (1/3)h(a₁² + a₁a₂ + a₂²). Top side 4 cm, bottom 8 cm, height 6 cm gives 224 cm³.
Converting Between Cubic Units and Liquid Volume
Metric Liquid Volumes
The metric system makes this easy. 1 liter equals exactly 1,000 cm³—they defined it that way on purpose. One milliliter = one cubic centimeter. No math needed.
A cubic meter holds 1,000 liters. That's a big volume—think swimming pools, storage tanks, room capacities. 1 m³ = 1,000 L = 1,000,000 mL = 1,000,000 cm³.
A 10 × 10 × 10 cm box holds exactly 1 liter (1,000 cm³). Your typical 500 mL water bottle? That's 500 cm³ of space. A 2 m³ storage tank fits 2,000 liters.
US Liquid Volumes
The US gallon is roughly 3.8 liters (3,785 cm³). That's what you'll see on milk jugs, gas pumps, water bottles—basically any liquid volume in America. It breaks down into 4 quarts, 8 pints, 16 cups, or 128 fluid ounces.
Fluid ounces are small: about 30 mL each. Eight of them make a cup (237 mL). Standard US cooking measurements.
A gallon milk jug holds around 3,800 mL. A 2-liter soda bottle is about half a gallon. Your 8 oz coffee cup? That's roughly 240 mL.
Imperial (UK) Volumes
1 Imperial gallon ≈ 4.546 liters. This is different from the US gallon. The UK gallon is larger. It's used in the UK, Canada (in older contexts), and some Commonwealth countries.
UK gallons and US gallons are different—don't mix them up. One UK gallon equals about 1.2 US gallons. When you see "gallon" in a measurement, check whether it's US or Imperial to avoid mistakes.
Cubic Units to Liquid Conversions
Converting between cubic units and liquid volumes is straightforward. Cubic centimeters to liters? Divide by 1,000 (so 5,000 cm³ = 5 L). Cubic meters to liters? Multiply by 1,000 (0.5 m³ = 500 L).
For imperial conversions, 231 cubic inches equals exactly one US gallon. That means 1,000 in³ comes out to about 4.33 gallons. Cubic feet are easier—just multiply by 7.48. A 100 ft³ pool holds roughly 748 gallons.
Fish tank measuring 50 × 30 × 40 cm? That's 60,000 cm³, which works out to 60 liters or roughly 16 gallons. Pool that's 10 × 20 × 5 feet comes to 1,000 ft³, about 7,500 gallons or 28,000 liters.
Why Different Systems?
Cubic units (cm³, m³, in³, ft³) are used in science, engineering, and construction. They're universal and based on length cubed. Liquid volumes (liters, gallons, mL) are used in cooking, beverages, and everyday life. They're more convenient for pouring and measuring liquids. It's the same measurement—just different units for different contexts. Knowing the conversions lets you work in both systems.
Converting Volume Units
Metric Volume Conversions
1 m³ = 1,000,000 cm³. Why? Because 1 m = 100 cm. So 1 m³ = 100 cm × 100 cm × 100 cm = 1,000,000 cm³. You must cube the conversion factor!
1 cm³ = 1,000 mm³. Since 1 cm = 10 mm, we get 1 cm³ = 10³ mm³ = 1,000 mm³.
1 km³ = 1,000,000,000 m³. Since 1 km = 1,000 m, we get 1 km³ = 1,000³ m³ = 1 billion m³.
Converting 5 m³ to cm³? Multiply by a million: 5,000,000 cm³. Going the other way, 250,000 cm³ ÷ 1,000,000 = 0.25 m³.
Imperial Volume Conversions
1 ft³ = 1,728 in³. Since 1 ft = 12 in, we get 1 ft³ = 12³ in³ = 1,728 in³.
1 yd³ = 27 ft³. Since 1 yd = 3 ft, we get 1 yd³ = 3³ ft³ = 27 ft³.
Need 3 ft³ in cubic inches? That's 3 × 1,728 = 5,184 in³. Converting 54 ft³ to yards: 54 ÷ 27 = 2 yd³.
Metric to Imperial
1 in³ ≈ 16.387 cm³. Since 1 in = 2.54 cm, we get 1 in³ = 2.54³ ≈ 16.387 cm³.
1 ft³ ≈ 0.0283 m³ or ≈ 28,317 cm³. Since 1 ft = 0.3048 m, we get 1 ft³ = 0.3048³ ≈ 0.0283 m³.
1 m³ ≈ 35.315 ft³. This is the inverse of the above conversion.
1 cm³ ≈ 0.061 in³. This is the inverse of 16.387.
Construction Volume Units
Cubic yards (yd³) are the standard for concrete, mulch, soil, and gravel. Dump truck capacity is measured in cubic yards. Remember: 1 yd³ = 27 ft³.
Driveway that's 20 × 10 feet and 6 inches thick? That's 100 ft³, which converts to about 3.7 yd³. You'd order 4 cubic yards to be safe.
Why Cube the Conversion
Common mistake: 1 m = 100 cm, so people incorrectly think 1 m³ = 100 cm³. Wrong! Volume is three-dimensional. You must cube the factor: 1 m³ = 100 cm × 100 cm × 100 cm = 1,000,000 cm³. Always cube the linear conversion factor when converting volume. 1 ft = 12 in means 1 ft³ = 1,728 in³ (not 12). This is crucial for accurate volume conversions.
Quick conversions you'll use all the time: 1 m³ = 1,000 L = 264 gal (US) = 35.3 ft³. Going the other way, 1 ft³ = 7.48 gal (US) = 28.3 L. For smaller volumes: 1 L = 1,000 mL = 1,000 cm³ = 0.264 gal (US). And 1 gal (US) = 3.785 L = 3,785 mL = 231 in³.
Real-World Uses of Volume Calculations
Swimming Pools and Hot Tubs
Got a rectangular pool that's roughly 20 × 10 feet and averages 5 feet deep? That's about 1,000 cubic feet, which works out to around 7,500 gallons. You need this number for everything—chlorine dosing, filter sizing, heater selection. Get it wrong and you're either wasting chemicals or dealing with algae.
Round pools are trickier. A 15-foot diameter pool (7.5 ft radius) that's 4 feet deep? About 700 ft³ or 5,300 gallons. Always round up a bit for chemicals—better to have slightly too much capacity than too little.
Hot tubs are smaller but the same logic applies. A 6 × 6 foot tub that's 3 feet deep holds roughly 800 gallons. Most heaters need about 1.5 kW per 100 gallons, so that's a 12 kW heater minimum.
Construction and Concrete
Foundation slab for a small building—say 40 × 30 feet and 6 inches thick. That's 600 cubic feet of concrete, which converts to about 22 cubic yards (since there are 27 ft³ per yd³). You'd order 23 yards to be safe. At $120 per yard? You're looking at $2,800 or so.
Walkways are thinner. A 50-foot path that's 4 feet wide and 4 inches thick needs maybe 2.5 cubic yards. Order 3 to account for uneven ground and waste.
The trick with concrete is always ordering a bit extra—you can't run to the store if you're short, and having leftovers is way better than having a half-finished pour.
Shipping and Logistics
Carriers charge by dimensional weight, which is basically volume-based pricing. That 12 × 8 × 6 inch box? It's about 576 in³ or 0.33 ft³. If it's full of feathers (super light), you still pay based on the space it takes up in the truck.
Standard 20-foot shipping containers hold about 1,360 cubic feet on paper. In reality? You'll fit maybe 1,000 cubic feet of actual boxes because packing isn't perfect and odd shapes waste space.
Cooking and Baking
Recipe calls for a 9 × 13 inch pan but you've got an 8-inch square? The 9×13 is roughly double the volume (234 in³ vs 128 in³), so you need to scale the recipe by about 1.8x. Or just make two 8-inch pans worth.
For liquids, volume and mass are interchangeable with water: 1 mL = 1 gram. Two cups of water? That's 474 mL and weighs 474 grams. Flour's different (lighter), but water's a good baseline.
Aquariums and Tanks
A 50 × 30 × 40 cm fish tank holds 60 liters (about 16 gallons). The old "1 inch of fish per gallon" rule means you can stock maybe 15-16 small fish. But honestly, less is better—fish need space.
Filters? Get one rated for 2-3x your tank volume per hour. So a 60-liter tank wants a filter pumping 120-180 L/hour. Bigger is usually better here.
Weekly water changes: swap out 25% or so. For that 60L tank, that's 15 liters. Make sure you dechlorinate it before adding.
Fuel and Storage Tanks
Horizontal cylindrical fuel tank, maybe 2 feet in radius and 6 feet long? That's about 75 cubic feet, or roughly 560 gallons. Common for home heating oil or propane.
Big propane tanks are often spherical. A 5-foot radius sphere holds around 520 ft³—that's nearly 4,000 gallons. Spheres are structurally efficient for pressurized storage.
Rain barrels are usually 50-70 gallons. A cylinder that's 1 foot radius and 3 feet tall gives you about 70 gallons—not much, but it adds up during a storm.
Medicine and Healthcare
Your body is mostly water. An average 70 kg (154 lb) adult contains roughly 70 liters of fluid. That's not trivial—medication dosing accounts for this, especially with IV drugs.
IV bags come in 500 mL or 1 L sizes. Drip rates are set in mL/hour depending on what the patient needs. Too fast and you overload the system; too slow and you under-treat.
Your lungs hold about 6 liters total, but you only use about 500 mL per normal breath (tidal volume). Deep breaths can hit 3-4 liters if you really push it.
Volume, Mass, and Density Relationship
What Is Density?
Density measures mass per unit volume—how much "stuff" is packed into a space. The formula is ρ = m/V (rho = mass divided by volume). Density is measured in g/cm³, kg/m³, or lb/ft³. Dense materials like lead and gold have high mass in small volume. Light materials like foam and air have low mass in large volume. Density determines whether objects float or sink.
Calculating Mass from Volume
Formula: mass = density × volume. If you know volume and density, you can find mass.
Take a 10 cm gold cube—that's 1,000 cm³ of space. Gold's density is 19.3 g/cm³, so the mass works out to 19,300 grams (19.3 kg). That's over 40 pounds for something the size of a Rubik's cube. Gold is ridiculously heavy.
Same cube, but styrofoam instead? Styrofoam's only about 0.05 g/cm³, so you get 50 grams total. Light enough to toss around.
Water's the easy one to remember: 1 g/cm³ means 1 cm³ = 1 gram exactly. A cubic meter of water (that's 1,000 liters) weighs exactly 1,000 kilograms—one metric ton.
Calculating Volume from Mass
Formula: volume = mass / density. If you know mass and density, you can find volume.
Say you've got 500 kg of steel. Steel runs about 7,850 kg/m³. Divide mass by density: 500 ÷ 7,850 gets you roughly 0.064 m³, which is about 64 liters.
Common Densities
Water is easy—1 g/cm³, which means one liter weighs exactly one kilogram. Ice is less dense at 0.92 g/cm³, which is why it floats. Air is incredibly light at 0.0012 g/cm³ at sea level. Helium's even lighter (0.00017 g/cm³), hence balloons.
Metals get heavy fast. Gold tops the common metals at 19.3 g/cm³—nearly 20 times denser than water. Lead comes in at 11.34 g/cm³, iron at 7.87 g/cm³, and aluminum is the lightweight of the group at 2.70 g/cm³. Mercury's weird because it's liquid but still denser than most solid metals (13.53 g/cm³).
Most woods float because they're less dense than water. Oak is around 0.75 g/cm³, pine is about 0.5 g/cm³, and balsa is as low as 0.15 g/cm³. Concrete sits around 2.3 g/cm³, glass around 2.5 g/cm³, and plastics range from 0.9 to 1.4 g/cm³ depending on the type.
Buoyancy and Floating
Archimedes' Principle: An object floats if its density is less than the fluid density. It sinks if density is greater than the fluid.
Wood with density 0.6 g/cm³ floats in water (1.0 g/cm³). Iron at 7.87 g/cm³ sinks. Ice sits at 0.92 g/cm³, which means it floats but stays mostly underwater—about 92% submerged, 8% sticking up. That's why icebergs are dangerous.
Boat paradox: A steel ship floats despite steel being denser than water. Why? The volume includes air inside the hull. The overall average density (steel + air) is less than water. Shape matters for buoyancy!
Volume Calculation Tips and Common Mistakes
How to Get Accurate Measurements
Measure carefully. Here's the thing: volume magnifies your errors. A tiny 1% mistake in length becomes a 3% mistake in volume (because you're cubing it). Use a good tape measure, calipers, or even a laser measurer—it's worth it.
Keep your units consistent. Mixing meters and centimeters is asking for trouble. Got 5 m × 200 cm × 30 cm? Convert everything first: 5 m × 2 m × 0.3 m = 3 m³. Do the conversion up front, not halfway through when you're already confused.
Know your shape. Sounds obvious, but you'd be surprised. Is it actually a perfect cylinder or is it slightly tapered? A true sphere or more egg-shaped? Use the right formula for what you've actually got.
Break weird shapes into simple ones. Got an L-shaped pool? That's just two rectangles. Calculate each part, add them up. No special formula needed—just divide and conquer.
Always add a buffer for construction. Ordering concrete? Add 10% for waste, spillage, and uneven ground. You can't call the truck back if you're short, and leftover concrete is way better than a half-finished job.
Don't pretend you're more precise than you are. If you measured with a tape measure in feet, don't write down 123.45678 ft³. Round to 123.5 or even 124. False precision just looks silly.
Sanity check your answers. A bedroom with 500 m³ volume? That's warehouse-sized—definitely wrong. Quick reasonableness checks catch most calculation errors before you order materials or fail the homework.
Common Screw-Ups (and How to Avoid Them)
Mixing up volume and area. Volume = cubic units (cm³, ft³). Area = square units (cm², ft²). They're not interchangeable. A box has volume; a flat square has area. Different formulas, different purposes.
Another thing—forgetting to write units. "The volume is 50" is meaningless. 50 what? Teaspoons? Cubic kilometers? Always include units or nobody knows what you're talking about.
You have to cube the conversion factor when dealing with volume. 1 meter = 100 centimeters, sure. But 1 m³ = 1,000,000 cm³ (that's 100 × 100 × 100). Skip this and you're off by factors of 100 or 1,000.
Diameter vs. radius mistake: This one catches everyone. The formulas want radius, not diameter. If your sphere has a 10 cm diameter, you need to use r = 5 cm in the formula. Using 10 instead of 5 doesn't just double your answer—it makes it eight times bigger. Don't do that.
Cone volume isn't πr²h—it's (1/3)πr²h. That division by 3 is critical. A cone is exactly one-third the volume of a matching cylinder. Forget that fraction and your answer triples.
Slant height vs. vertical height—pyramids and cones need the straight-up-and-down height, not the slanted edge. Measure perpendicular from the base to the tip.
Wrong formula for the shape? Hemisphere isn't the same as a sphere—it's half, so the formula is different: (2/3)πr³ instead of (4/3)πr³. Check which shape you actually have.
Adding volumes without matching units. 5 m³ + 200 cm³ ≠ 5,200 anything. Convert first: 200 cm³ = 0.0002 m³, then add: 5.0002 m³. Always match units before combining.
Container volume vs. material volume—these are different things. A plastic tank might hold 100 liters inside, but the plastic itself only uses maybe half a liter. Interior capacity and material volume aren't the same.
Here's how to avoid screwing up: write the formula first, substitute your numbers second, then calculate. Doing it in that order means you catch stupid mistakes before they propagate through the whole problem and give you nonsense answers.