Ratio of Volumes

Count the cubes

A cube with edges 2 units long holds 2 × 2 × 2 = 8 unit cubes: 2 layers, each of 2 rows of 2.

A unit cube beside a cube of side 2, which is ruled into unit cubes on every face in view.

Double the edges

Double every edge, to 4 units. The cube now holds 4 × 4 × 4 = 64 unit cubes, and 64 ÷ 8 = 8. Doubling the lengths multiplied the volume by 8, not by 2.

Count it in copies of the smaller cube instead. The large cube is 2 small cubes wide, 2 deep and 2 tall, so it holds 2 × 2 × 2 = 8 of them. Each of the three directions multiplies the count by 2 once, so the volume is multiplied by $2^3=8$2³ = 8.

Here each small cube stands for the whole cube of side 2. The cube of side 4 is 2 of them wide, 2 deep and 2 tall: 8 copies, each holding 8 unit cubes, and 8 × 8 = 64.

Cubed for volume

The same count works for any scale factor k. A solid enlarged by k is k times as wide, k times as deep and k times as tall, so its volume is multiplied by $k\times k\times k=k^3$k × k × k = k³. Lengths have one dimension and are multiplied by k. Areas have two and are multiplied by $k^2$k². Volumes have three and are multiplied by $k^3$k³.

The volume formulas agree. A cuboid 1 by 2 by 3 has a volume of 6 cubic units. Enlarged by 2 it is 2 by 4 by 6, with a volume of 48, and 48 = 6 × 8. A cylinder has volume $\pi r^{2}h$πr²h; a cylinder with twice the radius and twice the height has volume $\pi \times 2r\times 2r\times 2h=8\pi r^{2}h$π × 2r × 2r × 2h = 8πr²h.

The rule works for scale factors below 1 as well. A model at half scale has every length halved, every area divided by 4 and every volume divided by 8, because ${\left(\frac{1}{2}\right)}^3=\frac{1}{8}$(1/2)³ = 1/8.

Lengths, areas and volumes side by side

Take two similar cylinders: one with a radius of 3 cm and a height of 5 cm, and one with a radius of 6 cm and a height of 10 cm. The scale factor is 6 ÷ 3 = 2.

A length: the circumference of the base, $2\pi r$2πr, is $6\pi$6π cm on the small cylinder and $12\pi$12π cm on the large one. That is 2 times as long.

An area: the curved surface, $2\pi rh$2πrh, is $2\pi \times 3\times 5=30\pi$2π × 3 × 5 = 30π cm² on the small cylinder and $2\pi \times 6\times 10=120\pi$2π × 6 × 10 = 120π cm² on the large one. That is 4 times as much, and $2^2=4$2² = 4.

A volume: $\pi r^{2}h$πr²h is $\pi \times 9\times 5=45\pi$π × 9 × 5 = 45π cm³ on the small cylinder and $\pi \times 36\times 10=360\pi$π × 36 × 10 = 360π cm³ on the large one. That is 8 times as much, and $2^3=8$2³ = 8.

Drag the corner of the cube to change the scale factor k. The line is k units long, the square holds k rows of k unit squares, and the cube holds k layers of k × k unit cubes. Make the area scale factor 9, and read the volume scale factor at the same k: 27.

Mass and capacity

The mass of a solid object made of one material is proportional to its volume: twice as much of the material has twice the mass. So similar solid objects made of the same material have masses in the ratio of their volumes, and the mass is multiplied by $k^3$k³.

A solid metal model of a statue is 20 cm tall and has a mass of 0.4 kg. The statue, made of the same metal, is 1 m tall, which is 100 cm. The scale factor is 100 ÷ 20 = 5, so the statue's mass is $0.4\times 5^3=0.4\times 125=50$0.4 × 5³ = 0.4 × 125 = 50 kg.

Capacity, the amount a container holds, is a volume too. A bowl holds 0.5 liters, and a similar bowl is 3 times as wide. It holds $0.5\times 3^3=0.5\times 27=13.5$0.5 × 3³ = 0.5 × 27 = 13.5 liters.

Working back with a cube root

Given the volumes of two similar solids, take the cube root of their ratio to find the ratio of their lengths. Two similar tins hold 128 cm³ and 250 cm³. Simplify the ratio first by dividing both numbers by 2: 128 : 250 = 64 : 125. The cube root of 64 is 4, because 4 × 4 × 4 = 64, and the cube root of 125 is 5, because 5 × 5 × 5 = 125. So the lengths are in the ratio 4 : 5, and the scale factor from the small tin to the large one is $\frac{5}{4}=1.25$5/4 = 1.25.

Once the lengths are known, the areas follow: they are in the ratio 16 : 25, since $4^2=16$4² = 16 and $5^2=25$5² = 25. If the label on the small tin has an area of 48 cm², the label on the large tin has an area of $48\times \frac{25}{16}=75$48 × 25/16 = 75 cm².

The usual mistakes

Multiplying the volume by k. A solid twice as tall is also twice as wide and twice as deep, so it holds 2 × 2 × 2 = 8 times as much, not 2 times as much.

Using $k^2$k², the rule for area. With a scale factor of 2, the label on a bottle is multiplied by $2^2=4$2² = 4, but the juice inside it fills a volume, so it is multiplied by $2^3=8$2³ = 8.

Going back with a square root. Volumes in the ratio 1 : 8 come from lengths in the ratio 1 : 2, because $2^3=8$2³ = 8. The square root belongs to areas.