How do I work out the volume of an odd shape?

Cut it into pieces that are each one of these shapes, work them out separately and add them up. An L-shaped room is two boxes; a silo is a cylinder plus a half-sphere; a bottle is often a cylinder plus a cone. Subtraction works too: a pipe is one cylinder with a smaller one removed from the middle.

Why is a cone exactly a third of a cylinder?

With the same base and the same height, it always is - and the same third appears between a pyramid and its box. You can see it with water: filling a cone and pouring it into a matching cylinder takes exactly three goes. The proof needs calculus, but the fact is old enough that Archimedes had it, and it is worth knowing because it turns two formulas into one.

Does the surface area include the base?

Yes - every figure here is closed, so the total covers all faces including bases and lids. If you are painting a tank that sits on the ground, or lining a box that stays open at the top, subtract the face you are not covering. That is one multiplication, and it beats a tool that guesses which faces you meant.

Can I use this for a swimming pool or aquarium?

Yes, with one adjustment: measure the water, not the container. A tank filled to five centimetres below the rim holds noticeably less than its full depth suggests, and a pool with a sloping floor is closer to two shapes than one. Working in centimetres makes the litre line directly useful, which is usually the number you actually wanted.

Volume Calculator - Cube, Box, Cylinder, Sphere, Cone and More

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Volume and surface area

Shape

Seven shapes, one habit of measuring

Volume questions arrive disguised as practical ones. How much soil fills that raised bed, how much water is in the pond, will this parcel go through as a small package, how much concrete for the footing. Each of them is a shape and a few measurements, and the arithmetic is old and settled. What is new is only that you no longer have to remember which formula belongs to which shape.

Using it in three steps

1 Pick the shape closest to the thing you are measuring.
2 Enter the measurements in one unit and stick to it - all of them in centimetres, or all in inches.
3 Read the volume in cubed units. The litre line applies only if you measured in centimetres.

Why doubling a size does not double the volume

Volume grows with the cube of size. Double every dimension of a box and it holds eight times as much, not twice. This is why a bin that looks only a little bigger swallows a great deal more, why a small increase in tank radius adds surprising capacity, and why shipping costs jump so sharply with parcel dimensions. Surface area meanwhile grows with the square, which is why bigger things are relatively cheaper to wrap and slower to cool.

Why the tool never asks for your units

Every formula here is dimensionally consistent, which means the units come out of the arithmetic on their own. Measure in centimetres and the volume is cubic centimetres; measure in inches and it is cubic inches. The only thing that must not happen is mixing: a width in centimetres with a height in inches produces a number that is not wrong so much as meaningless. The litre line is offered only for centimetres, because that is the one conversion that is exactly true.

Volume - frequently asked questions

Whichever you type in. The formulas do not know or care - they return the cube of whatever unit you used. Keep every field in the same unit and the answer is correct in that unit cubed. The litre figure appears because a cubic centimetre is exactly one millilitre, which makes that particular conversion free of rounding.

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