★ MATH · GEOMETRY · VOLUME
A cylinder with radius 3 and height 10 has a volume of 282.743339 — π times the radius squared times the height, rounded to 6 decimal places. Pick a shape and the calculator works out the volume: boxes, cylinders, spheres and cones, with the formula, the step-by-step breakdown and the answer to six places where π is involved. This one is the VOLUME tool — how much a shape holds — the area-calculator page is the AREA tool — the space on a flat surface — not a volume; the circumference-calculator page is the CIRCUMFERENCE tool — the distance around a circle — not a volume; the area-of-hemisphere-calculator page is the HEMISPHERE tool — the surface area and volume of a half-sphere — not the four-shape volume tool; and the box-size-chart page is the BOX SIZE chart — moving-box dimensions — not a volume calculator. No volume calculator exists on this site — this one is the first.
Volume is how much a shape holds: length × width × height for a box, π × radius² × height for a cylinder, 4/3 × π × radius³ for a sphere, and a third of that cylinder for a cone. On the defaults it is a cylinder of radius 3 and height 10, and π × 9 × 10 is 282.743339 to six places. Boxes are exact; the π shapes are rounded to 6 places because π never terminates.
It means a cylinder of radius 3 and height 10 holds 282.743339 cubic units — about 283 of them, if you were filling it with unit cubes and counting. The volume is the area of the circular base, π × 9, which is 28.274334, times the height, 10 — and 28.274334 × 10 is 282.743339. If the units were centimetres, that is 282.743339 cubic centimetres, or about 0.283 litres; if they were feet, it is 282.743339 cubic feet, or about 2,115 gallons. The number is the same either way — only the unit after it changes.
The box is exact; the π shapes are rounded. A box multiplies three lengths, and nothing in the multiplication rounds: a box 2 × 3 × 4 has a volume of exactly 24, and a box 12.5 × 7.25 × 2 is exactly 181.25. A cylinder, a sphere and a cone all multiply by π, which never terminates, so their answers stop at six places: a cylinder of radius 3 and height 10 is 282.743339, and the true value continues past the sixth decimal. Carry the 6-place value into further working only if that precision is enough.
A cone holds exactly one-third of its cylinder. A cone and a cylinder with the same radius and the same height are related by a fixed third: the cone of radius 3 and height 10 holds 94.247780, and the cylinder holds 282.743339 — three times as much, because 3 × 94.247780 is 282.743339. The sphere sits between them at 113.097336 for radius 3 — more than the cone, less than half the cylinder. If a cone ever comes out bigger than its cylinder, the numbers are wrong.
The radius is squared — small errors grow. The cylinder and the cone both square the radius, so a radius that is off by a little makes the volume off by about twice as much in proportion: a radius of 3.03 instead of 3 is 1% too big, and the volume comes out about 2% too big. The sphere cubes the radius, so the same 1% error makes the volume about 3% too big. Measure the radius twice; the height only needs measuring once.
On the defaults: a cylinder of radius 3 and height 10.
And the other way: a cylinder of height 10 with a volume of 282.743338 or more needs a radius of 3 — the reverse box finds the radius from the volume.
The defaults are a cylinder of radius 3 and height 10, chosen so the answer is a clean multiple of π: 90π is 282.743339. They are a starting point, not a lesson: your shape may be any of the four, and the calculator shows the formula it used, the step-by-step breakdown, and the verdict. Lengths must be more than 0 — a radius of 0 is not a shape — and the π shapes are rounded to 6 decimal places because π never terminates. Put your numbers in, read the answer, and check the working before you use it.
These notes repeat in every explainer section because this page is built to be read start to finish: the box is exact and the π shapes are rounded; a cone holds exactly one-third of its cylinder; and the radius is squared, so small errors grow.
| Output | Formula | What it means |
|---|---|---|
| Shape | input | Cylinder, radius 3, height 10. |
| Volume | V = π × radius² × height | 282.743339. |
| Volume (6dp unrounded) | decimal, 6 places | 282.743339. |
| Formula | by shape | V = π × radius² × height. |
| Breakdown | step by step | π × 3² × 10 = 282.743339. |
| Verdict | in words | A cylinder with radius 3 and height 10 has a volume of 282.743339. |
| Reverse: Smallest whole height for a 2 × 3 box volume of 24 or more | step | 4: Box 2 × 3 × 4: volume 24. |
| Reverse: Smallest whole radius for a cylinder (height 10) volume of 282.743338 or more | step | 3: Cylinder, radius 3, height 10: volume 282.743339. |
| Reverse: Smallest whole radius for a sphere volume of 113.097335 or more | step | 3: Sphere, radius 3: volume 113.097336. |
| Reverse: Smallest whole height for a cone (radius 3) volume of 94.247779 or more | step | 10: Cone, radius 3, height 10: volume 94.247780. |
| Reverse: Largest whole radius for a sphere volume of 113.097336 or less | step | 3: Sphere, radius 3: volume 113.097336. |
| Reverse: Smallest whole radius for a cylinder (height 5) volume of 392.699081 or more | step | 5: Cylinder, radius 5, height 5: volume 392.699082. |
| Cylinder, radius 3, height 10 | same model | Cylinder, radius 3, height 10: volume 282.743339. |
| Box 2 × 3 × 4 | same model | Box 2 × 3 × 4: volume 24. |
| Sphere, radius 3 | same model | Sphere, radius 3: volume 113.097336. |
| Cone, radius 3, height 10 | same model | Cone, radius 3, height 10: volume 94.247780. |
| Box 12.5 × 7.25 × 2 | same model | Box 12.5 × 7.25 × 2: volume 181.25. |
| Cylinder, radius 5, height 5 | same model | Cylinder, radius 5, height 5: volume 392.699082. |
| Sphere, radius 1 | same model | Sphere, radius 1: volume 4.188790. |
| Cone, radius 2, height 6 | same model | Cone, radius 2, height 6: volume 25.132741. |
| Box 10 × 10 × 10 | same model | Box 10 × 10 × 10: volume 1000. |
| Sphere, radius 5 | same model | Sphere, radius 5: volume 523.598776. |
| Box 0 × 3 × 4 (zero length) | same model | Lengths must be more than 0.. |
| Cylinder, radius −3 (negative) | same model | Lengths must be more than 0.. |
| Non-numeric input | same model | Enter a number for the length, the width and the height.. |
| Sphere, radius 0 | same model | Lengths must be more than 0.. |
| Cone, height 0 | same model | Lengths must be more than 0.. |
| Box 0.5 × 0.25 × 2 | same model | Box 0.5 × 0.25 × 2: volume 0.25. |
| Cylinder, radius 1, height 1 | same model | Cylinder, radius 1, height 1: volume 3.141593. |
| Sphere, radius 2.5 | same model | Sphere, radius 2.5: volume 65.449847. |
| Cone, radius 1, height 1 | same model | Cone, radius 1, height 1: volume 1.047198. |
| Box 100 × 100 × 100 | same model | Box 100 × 100 × 100: volume 1000000. |
| Cylinder, radius 10, height 10 | same model | Cylinder, radius 10, height 10: volume 3141.592654. |
The arithmetic is exact on the numbers you type: a box 2 × 3 × 4 has a volume of exactly 24, and a box 12.5 × 7.25 × 2 is exactly 181.25. Cylinder, sphere and cone results use π and are rounded to 6 decimal places; the true values continue past the sixth decimal, so carry the 6-place value into further working only if that precision is enough. The calculator refuses zero or negative lengths and non-numeric input. All four shapes assume the measurements are of the shape itself — a tank’s wall thickness, a box’s lid, or a cone’s tip cut off are not modelled, so measure the inside if you want the capacity. Treat the breakdown as the check — if a length is wrong, the breakdown shows it.
A cylinder with radius 3 and height 10 has a volume of 282.743339 — π times the radius squared times the height, rounded to 6 decimal places. A box 2 × 3 × 4 has a volume of 24. A sphere with radius 3 has a volume of 113.097336. A cone with radius 3 and height 10 has a volume of 94.247780. Smallest whole height for a 2 × 3 box volume of 24 or more: 4. Smallest whole radius for a cylinder (height 10) volume of 282.743338 or more: 3. Smallest whole radius for a sphere volume of 113.097335 or more: 3. Smallest whole height for a cone (radius 3) volume of 94.247779 or more: 10. Largest whole radius for a sphere volume of 113.097336 or less: 3. Smallest whole radius for a cylinder (height 5) volume of 392.699081 or more: 5. Source: Bright Side Kit, brightoolbox.com.
The volume is 282.743339, because π × 3² × 10 is 282.743339. The circular base is π × 9, which is 28.274334, and 28.274334 × 10 is 282.743339 — rounded to 6 decimal places because π never terminates.
Multiply length × width × height — a box 2 × 3 × 4 has a volume of 24. The answer is exact: no π anywhere, so no rounding anywhere. A box 12.5 × 7.25 × 2 has a volume of 181.25.
Take four-thirds π times the radius cubed — a sphere of radius 3 has a volume of 113.097336. A sphere of radius 1 has a volume of 4.188790, and a sphere of radius 5 has a volume of 523.598776.
Take one-third π times the radius squared times the height — a cone of radius 3 and height 10 has a volume of 94.247780. A cone holds exactly one-third of the cylinder with the same radius and height: 3 × 94.247780 is 282.743339.
The volume is 392.699082, because π × 25 × 5 is 392.699082. That is 125π, rounded to 6 decimal places.
Undo the formula — a cylinder of height 10 with a volume of 282.743338 or more needs a radius of 3. A sphere with a volume of 113.097335 or more needs a radius of 3, and a cone of radius 3 with a volume of 94.247779 or more needs a height of 10.
Because a box multiplies three lengths and nothing in the multiplication rounds, while a cylinder multiplies by π, which never terminates. The display stops at six places for the π shapes; the box answer is the whole answer.
The volume is 1,000 — 10 × 10 × 10, exact. A cube is a box with all three sides equal, so the box formula does it: length × width × height.