Volume Calculator
Six shapes, volume and surface area for each. The units are whatever you type, cubed — so centimetres in gives cubic centimetres, which are millilitres.
Volume
3,000.0000
- Volume
- 3,000.0000
- Surface area
- 1,300.0000
- Formula
- length × width × height
- If in cm — litres
- 3.000
- If in inches — US gallons
- 12.987
- The units are whatever you typed, cubed. Enter centimetres and the volume is cubic centimetres, which is millilitres — so a thousand of them is a litre.
- Scaling matters more in three dimensions than people expect. Doubling every dimension multiplies the volume by eight and the surface area by four, which is why large animals overheat and small ones lose heat.
About volume calculations
A box is length times width times height, and everything else is a variation on measuring a cross-section and pushing it through a distance. A cylinder is a circle pushed through its height: πr² for the circle, times the height.
The cone and the pyramid share a factor of one third with their enclosing shapes, and it is exactly a third rather than approximately. A cone is a third of the cylinder that would contain it; a pyramid is a third of its box. That relationship is one of the tidier facts in solid geometry and worth remembering — it means you can work out either by doing the easy shape and dividing.
Scaling in three dimensions catches people out. Doubling every dimension multiplies the volume by eight and the surface area by only four. That difference is why large animals struggle to shed heat and small ones struggle to keep it, why a big saucepan takes so much longer to come to the boil, and why storage costs scale better than packaging does.
Cubic centimetres are millilitres, so a volume in centimetres divides by a thousand to give litres. Cubic inches divide by 231 to give US gallons. Both are shown because a volume question is usually really a question about how much something holds.
What it works out
- Cube, box, cylinder, sphere, cone and pyramid
- Volume and surface area for each
- Litres and gallons, if the input was cm or inches
The formula
Box: l × w × h Cylinder: πr²h Sphere: ⁴⁄₃πr³ Cone: ⅓πr²h
A 20 by 15 by 10 box is 3,000 cubic units. If those were centimetres, that is three litres.
The cylinder is a circle extruded: the area πr² times the height. A radius of 3 and a height of 5 gives 141.37.
The cone is exactly a third of that same cylinder — 47.12 — and the pyramid is exactly a third of its box. Not approximately a third; exactly. Both fall out of the same integration and it is worth knowing as a shortcut.
Surface area is a different calculation and it does not scale with volume. A cube of side 3 has a volume of 27 and a surface of 54. Double the side to 6 and the volume becomes 216 — eight times — while the surface becomes 216 as well, only four times what it was.
For the cone, the curved surface uses the slant height rather than the vertical one. A radius of 3 and a height of 4 gives a slant of 5, by Pythagoras, and a total surface of 75.40.
- r
- Radius, for round shapes.
- h
- Height — the vertical distance, not the slant.
- π
- About 3.14159.
- ⅓
- The factor that makes a cone from a cylinder, or a pyramid from a box.
A worked example
A box measuring 20 by 15 by 10.
That works out to 3,000.0000 .
- Volume
- 3,000.0000
- Surface area
- 1,300.0000
- Formula
- length × width × height
- If in cm — litres
- 3.000
- If in inches — US gallons
- 12.987
Questions
How do I calculate the volume of a cylinder?
Multiply the area of the circular end by the height — πr²h. A radius of 3 and a height of 5 gives 141.37.
Is a cone really a third of a cylinder?
Exactly a third, given the same radius and height. The same holds for a pyramid against its box. It is a useful shortcut — work out the easy shape and divide.
What happens if I double every dimension?
The volume goes up eight times and the surface area only four. That mismatch is why large animals overheat, why a big pan takes so long to boil, and why bulk storage is cheap relative to packaging.
How do I convert cubic centimetres to litres?
Divide by a thousand. A cubic centimetre is a millilitre, so 3,000 of them is three litres. Cubic inches divide by 231 for US gallons.
What units does the answer use?
Whatever you typed, cubed. Enter metres and the volume is cubic metres; enter inches and it is cubic inches. Mixing units between the fields is the usual error.
Why does a cone's surface use the slant height?
Because the curved surface runs up the slope, not straight up the middle. The slant comes from Pythagoras on the radius and the vertical height — 3 and 4 give a slant of 5.
How do I find the volume of an irregular shape?
Split it into shapes you can calculate and add them. For something genuinely irregular, displacement works — submerge it and measure the water it pushes aside.