Scaling Area and Volume

Scale every length

To scale a shape by a scale factor k, multiply every length in it by k: every edge, every height, every width. The new shape is similar to the old one: it has the same angles and the same shape, and it is larger when k is more than 1 and smaller when k is less than 1.

Start with a cube whose edges are 1 unit long. Each face is a unit square, with an area of 1 square unit, and the whole cube has a volume of 1 cubic unit. Now double every length, so each edge is 2 units long. The front face is now 2 units wide and 2 units tall, so it holds 2 rows of 2 small squares: 2 × 2 = 4 of them. Doubling the lengths multiplied the area of the face by 4, not by 2.

Every edge of the cube on the left is doubled. The front face of the large cube holds 2 rows of 2 unit squares, which is 4 of them.

A layer for every unit of depth

The front face is only the first layer. The large cube is also 2 units deep, so a second layer of 4 cubes sits behind the first. That makes 2 × 4 = 8 unit cubes, so the volume is multiplied by 8.

The large cube is 2 layers deep, and each layer holds 4 unit cubes: 2 × 4 = 8.

Squared for area, cubed for volume

Triple every length instead. Each face is now 3 by 3, so it holds 3 × 3 = 9 unit squares. The cube is 3 layers deep, so it holds 3 × 9 = 27 unit cubes.

The same happens for every scale factor. An area is two lengths multiplied together, and each length is multiplied by k, so the area is multiplied by $k\times k=k^2$k × k = k². A volume is three lengths multiplied together, so it is multiplied by $k\times k\times k=k^3$k × k × k = k³. Multiplying every length by 4 multiplies every area by 16 and every volume by 64.

The rule is not only for cubes. Any solid can be filled with small cubes, as closely as you like if the cubes are small enough. Scaling the solid by k scales each small cube into a cube k times as long, which holds $k^3$k³ times as much. Every piece grows by $k^3$k³, so the whole solid does too. In the same way, each small square on its surface grows by $k^2$k², so its surface area, the total area of its outside, grows by $k^2$k².

Tripled, each face holds 3 × 3 = 9 unit squares, and the cube holds 3 layers of 9, which is 27 unit cubes.

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.

The surface grows more slowly than the volume

Because the surface area grows by $k^2$k² and the volume by $k^3$k³, the larger solid has less surface for each unit of volume. The cube with edges of 1 unit has six faces of 1 square unit, so its surface area is 6 and its volume is 1: 6 square units of surface for each cubic unit. Double it, and the surface area is 6 × 4 = 24 while the volume is 8, which is 24 ÷ 8 = 3 square units for each cubic unit, half as much.

In general, surface area ÷ volume is multiplied by $k^2/k^3=\frac{1}{k}$k² / k³ = 1/k. Tripling the lengths leaves a third as much surface for each unit of volume.

Going back from a ratio

Given the ratio of two volumes of similar solids, take the cube root to find the ratio of their lengths. Two similar jugs hold volumes in the ratio 8 : 27. The cube root of 8 is 2 and the cube root of 27 is 3, so their heights are in the ratio 2 : 3. Their surface areas are then in the ratio $2^2$2² : $3^2=4$3² = 4 : 9.

Given the ratio of two areas, take the square root instead. Two similar boxes whose faces have areas in the ratio 9 : 25 have lengths in the ratio 3 : 5, so their volumes are in the ratio 27 : 125.

Two slips

Multiplying the volume by k alone. A cube with every edge doubled is twice as wide, but it is also twice as tall and twice as deep, so it holds 2 × 2 × 2 = 8 times as much, not 2 times as much.

Using $k^2$k² for a volume. The area factor, 2 × 2 = 4, counts one layer of the doubled cube. A volume has a third direction, so there are 2 layers: 2 × 4 = 8.