Free Volume Calculator with Steps
Pick a solid — cube, rectangular prism, sphere, cylinder, cone, or square pyramid — enter the measurements you have, and get the volume with the formula, the substitution, and the result written out line by line. Answers that involve π are shown in exact form first, so you can copy the working, not just the number.
Pick a solid and enter its measurements
The formula, substitution, and result update as you type. All measurements must use the same unit.
Passed through and cubed — "cm" gives cm³. No conversion.
Cube volume
Lengths in cm multiply three times, so the volume is in cm³.
V = 125 cm³
Step-by-step solution
Step 1 · Write the formula
A cube is a prism whose base area (a²) and height (a) use the same edge, so base × height collapses to a³.
V = a³
Step 2 · Substitute the measurements
Replace each symbol with its value.
V = 5³ = 5 × 5 × 5
Step 3 · Evaluate
Multiply the numbers through. The unit multiplies three times too, which is why the answer is cubed.
V = 125 cm³
The six volume formulas, and where they come from
Every solid in this online volume calculator uses one of six formulas, and four of them are really the same idea: volume = base area × height. A rectangular prism has a base of area l × w, so V = l × w × h. A cube is the special case where every edge matches, so V = a × a × a = a³. A cylinder has a circular base of area πr², so V = πr²h. If you can find the area of the base, you already know most of the volume formula.
The cone and the square pyramid each hold exactly one third of that base-times-height product: V = (1/3)πr²h for the cone and V = (1/3)a²h for the pyramid. The 1/3 is not an approximation — a cone contains exactly one third of the cylinder that shares its base and height, a fact you can verify physically by filling a cone with water and pouring it into the matching cylinder three times.
The sphere is the odd one out because it has no flat base: V = (4/3)πr³. Archimedes proved that a sphere fills exactly two thirds of the tightest cylinder that fits around it — that cylinder has volume πr² × 2r = 2πr³, and two thirds of 2πr³ is (4/3)πr³. He considered it his finest result and asked for the sphere-and-cylinder figure to be carved on his tomb.
Worked examples: cylinder and sphere volume by hand
Cylinder: suppose a can has radius r = 4 cm and height h = 10 cm — the same numbers pre-loaded when you pick the cylinder above, so this page doubles as a cylinder volume calculator you can check line by line. Write V = πr²h, substitute to get V = π × 4² × 10, square the radius first to get π × 16 × 10, and multiply: V = 160π ≈ 502.65 cm³. Notice that only the radius is squared — the height is multiplied in once, unsquared.
Sphere: pick the sphere and the same tool becomes a sphere volume calculator. With radius r = 3 cm, write V = (4/3)πr³, substitute to get (4/3) × π × 3³ = (4/3) × π × 27, and simplify: V = 36π ≈ 113.1 cm³. Many teachers want that exact form, 36π, rather than a rounded decimal — it is the answer with zero rounding error, which is why the calculator prints the π form on its own line before the decimal.
Units: why the answer is always cubed
Volume multiplies three lengths together, so the units multiply too: cm × cm × cm = cm³. The unit box in the calculator is a label that gets passed through and cubed — enter measurements in inches and the answer is in in³. It never converts anything, so if your measurements are mixed (a height in meters, a radius in centimeters), convert them to one unit before you type them in.
Two conversions are worth memorizing. First, 1 cm³ is exactly 1 mL, so a volume in cm³ is already a capacity in milliliters — the 160π ≈ 502.65 cm³ can above holds about half a liter. Second, converting cubed units means cubing the conversion factor: 1 m = 100 cm, but 1 m³ = 100³ = 1,000,000 cm³. Multiplying by 100 instead of 1,000,000 is one of the most common unit errors in volume homework.
Common mistakes, and how to check your answer
To sanity-check any answer, box the solid in: a solid can never have more volume than the smallest rectangular box it fits inside. A cylinder fills about 78.5% of its box (π/4 of it), a sphere fills about 52.4% of the tightest cube around it (π/6), and a cone is exactly one third of its cylinder. Also remember that volume scales with the cube of length: doubling every measurement multiplies the volume by 8, so a modest change in an input legitimately produces a large change in the answer — that is the cubing, not a mistake.
- Entering the diameter as the radius. This is the single most common volume error: it inflates a cylinder or cone by 4 times and a sphere by 8 times, because the mistake gets squared or cubed. If you measured across the full circle, halve it first.
- Using the slant height of a cone or pyramid. The h in every formula here is the perpendicular height, straight from the center of the base up to the apex. The slant height along the sloped side is always longer, and it belongs to surface-area formulas, not volume.
- Dropping the 1/3 on cones and pyramids — that computes the surrounding cylinder or prism instead, three times too big.
- Squaring the wrong thing in πr²h: only the radius is squared, never the height.
- Rounding π to 3.14 in the middle of a calculation. Keep π symbolic until the final line — the calculator shows the exact π form so you round once, at the end.
Frequently Asked Questions
Do the formulas use radius or diameter?
Radius — the distance from the center to the edge, which is half the diameter. If a problem gives the diameter of a sphere, cylinder, or cone, divide it by 2 before entering it. Using the full diameter makes a cylinder or cone come out 4 times too large and a sphere 8 times too large, because the error is squared or cubed along with the radius.
Why is there a 1/3 in the cone and pyramid formulas?
Because a cone or pyramid tapers from a full base to a single point, it holds exactly one third of the prism or cylinder that shares its base and height. That is an exact geometric fact, not a rounding: three cones of water fill the matching cylinder to the brim. Any solid that shrinks linearly from its base to an apex follows the same rule, V = (1/3) × base area × height.
What does the unit box do — does it convert units?
It is a label only, with no conversion. Whatever you type is attached to your measurements and cubed in the answer: cm gives cm³, in gives in³, m gives m³. Because nothing is converted, every measurement you enter must already be in that same unit — if a problem mixes meters and centimeters, convert them to one unit before entering them.
Which height do I enter for a cone or pyramid?
The perpendicular height: the straight distance from the center of the base up to the apex, meeting the base at a right angle. Do not use the slant height measured along the sloped side — it is always longer, and it is used for surface area, not volume. If you only know a cone's slant height ℓ and radius r, recover the height with the Pythagorean theorem: h = √(ℓ² − r²).
How precise is the π the calculator uses?
It computes with π at full double precision — about 16 significant digits — and rounds only the displayed result. For π-based shapes it also prints the exact form, like 160π, which has no rounding error at all. If your hand answer differs slightly from the calculator's, you most likely rounded π to 3.14 mid-calculation; that alone shifts 160π from 502.65 down to 502.4.
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