Volume

Area measures how much surface something covers; volume measures how much space something holds. How much water fits in a fish tank, how many blocks fill a box, how much drink is in a bottle — all of these are answered by the same kind of number: volume.

Volume is counted

To find how much space a box takes, the most direct method is to fill it with little cubes of edge 1 cm and count them: however many cubes it takes, that is the volume in cubic centimeters. It is the same idea as measuring area by covering a shape with unit squares — only now we fill instead of cover.

InteractiveVolume Cubes

Length l4Width w3Height h2

V = l × w × h = 4 × 3 × 2 = 24

Bottom layer

Layers

Watch the order of building: first lay the bottom layer — a base of length $l$ and width $w$ takes $l \times w$ cubes; then stack $h$ layers, which is $h$ copies of the base. So:

$V = l \times w \times h$

Forgot the formula? Rebuild the layers

Base area × height = volume. This "lay one layer, count the layers" idea works for the cuboid and for the cylinder too (base area $\pi r^2$ times height).

Units: cubic centimeters and liters

1 cm³: the size of a cube with 1 cm edges;
1 cm³ = 1 mL: that little cube holds exactly one milliliter of water;
1000 cm³ = 1 L: a liter is the capacity of a cube with edges of about 10 cm.

Liquids (milk, gasoline, medicine) are measured in L and mL; the space objects occupy is measured in cm³ and m³. Remember 1 L = 1000 mL and watch the orders of magnitude when converting.

Volume, area, perimeter: three siblings

The same idea of "size" gets a different name in one, two and three dimensions:

Perimeter: the length of one trip around the edge, in cm;
Area: the number of squares needed to cover the surface, in cm²;
Volume: the number of cubes needed to fill the space, in cm³.

One little cube of edge 1 cm: each edge is 1 cm long, each face is 1 cm², and the whole cube takes up 1 cm³ — three units for three dimensions.

Worked example: how much water fits the tank

The fish tank problem

Measured from the inside, a fish tank is 40 cm long, 30 cm wide and 25 cm high. How many liters does it hold?

$V = 40 \times 30 \times 25 = 30000 \ \mathrm{cm^3}$

Convert: 30000 cm³ = 30000 mL = 30 L. A small tank holds 30 liters — about 60 bottles of 500 mL water.

Convert carefully

1 L = 1000 mL, so 5000 cm³ is 5 L, not 50 L. Also, a real fish tank is never filled to the brim — the computed volume is the maximum it can hold, not the water actually inside.

Check yourself

Quick quiz

1. How much water fits at most in a 40 cm × 30 cm × 25 cm fish tank?
2. 1 cm³ equals how many milliliters?
3. What is the volume of a cube with edge 3 cm?