The Effect of Scale Factor on Volume

Q: How do I find the scale factor when I am only given two volumes?

Divide the larger volume by the smaller one to get $k^3$, then take the cube root. Volumes of $54$ and $2$ give $k^3 = 27$, so $k = 3$.

Double every dimension of a fish tank and it does not hold twice as much water. It holds eight times as much. The glass you need only goes up by a factor of four, and the edges only double. Three quantities, one scale factor, three different answers. That gap is where most scale factor mistakes live.

For two similar solids with scale factor $k$ , length is multiplied by $k$ , surface area by $k^2$ , and volume by $k^3$ . The exponent matches the number of dimensions the quantity measures. This article is about the volume case, the $k^3$ one, because it is the largest jump and the one students most often answer with a plain $k$ .

Why volume picks up a cube

Take a rectangular prism with length $\ell$ , width $w$ , and height $h$ , so its volume is $\ell w h$ . Now scale the solid by $k$ . Every length is multiplied by $k$ , so the new dimensions are $k\ell$ , $kw$ , and $kh$ , and the new volume is $(k\ell)(kw)(kh) = k^3 \ell w h$ . Three lengths each carry a factor of $k$ , and the three factors multiply into $k^3$ .

Nothing about that argument depends on the solid being a box. Volume is always a product of three lengths, whether it comes from $V = Bh$ , $V = \dfrac{1}{3}\pi r^2 h$ , or $V = \dfrac{4}{3}\pi r^3$ , so scaling the figure multiplies the volume by $k^3$ every time. A cone scaled by $k = 2$ holds $2^3 = 8$ times as much, and so does a sphere, a pyramid, or a grain silo.

Reading the scale factor off a ratio

Problems often hand you the two solids instead of $k$ . If a pair of corresponding lengths is given, the scale factor is their ratio: two similar cones with radii $2$ and $5$ have $k = \dfrac{5}{2}$ , so their volumes are in the ratio $5^3 : 2^3$ , which is $125 : 8$ . Cube each part of the length ratio and you have the volume ratio.

Going the other direction takes a cube root. If two similar solids have volumes $8$ and $216$ , the volume ratio is $\dfrac{216}{8} = 27$ , and the scale factor from the small solid to the large one is $\sqrt[3]{27} = 3$ . Once you have $k$ , every other comparison follows: surface areas here are in the ratio $3^2 : 1$ , which is $9 : 1$ .

The traps worth naming

The first trap is answering with $k$ where the question asks about volume. A solid scaled by $3$ has volume multiplied by $27$ , not by $3$ and not by $9$ . The wrong answers on a multiple-choice question are almost always $k$ , $k^2$ , and some multiple such as $3k$ , so decide which quantity is being measured before you compute anything.

The second trap is direction. Scaling up multiplies the volume by $k^3$ ; scaling down divides by $k^3$ . If a model is built at $k = \dfrac{1}{4}$ of full size, its volume is $\left(\dfrac{1}{4}\right)^3 = \dfrac{1}{64}$ of the real thing, which is why scale models look so much lighter than they seem like they should.

Worked examples

Example 1: scaling a volume up

A prism has volume $20$ . It is scaled by a factor of $k = 3$ . Find the new volume.

Volume is multiplied by the cube of the scale factor

k^3 = 3^3 = 27

Multiply the original volume

V = 20 \cdot 27

Simplify

V = 540

Answer: $540$ cubic units.

Example 2: a volume ratio from a length ratio

Two similar cones have radii $2$ and $5$ . Find the ratio of the larger volume to the smaller volume.

The scale factor is the ratio of corresponding lengths

k = \dfrac{5}{2}

Cube each part of the ratio

5^3 : 2^3

Simplify

125 : 8

Answer: $125 : 8$ .

Example 3: finding the scale factor from volumes

Two similar solids have volumes $8$ and $216$ . Find the scale factor from the small solid to the large one.

Form the volume ratio

\dfrac{216}{8} = 27

The volume ratio equals

k^3

k^3 = 27

Take the cube root

k = \sqrt[3]{27} = 3

Answer: $k = 3$ .

Example 4: surface area and volume in the same problem

A box-shaped tank measures $1$ ft by $3$ ft by $5$ ft, so its surface area is $46$ square feet and its volume is $15$ cubic feet. A similar tank is built with $k = 2$ . Find the new surface area and the new volume.

Surface area is multiplied by the square of the scale factor

2^2 = 4

New surface area

46 \cdot 4 = 184

Volume is multiplied by the cube of the scale factor

2^3 = 8

New volume

15 \cdot 8 = 120

Answer: $184$ square feet of surface area and $120$ cubic feet of volume.

Try one yourself

Common questions

If a solid is scaled by 3, why is the volume multiplied by 27 instead of 3?

Volume is a product of three lengths, and each of those lengths is multiplied by $3$ . That gives $3 \cdot 3 \cdot 3 = 27$ . Only a single length, such as an edge or a radius, is multiplied by $3$ alone.

How do I find the scale factor when I am only given two volumes?

Divide the larger volume by the smaller one to get $k^3$ , then take the cube root. Volumes of $54$ and $2$ give $k^3 = 27$ , so $k = 3$ .

Does surface area follow the same rule as volume?

No. Surface area is a two-dimensional measure, so it is multiplied by $k^2$ , while volume is three-dimensional and is multiplied by $k^3$ . Scaling by $2$ multiplies surface area by $4$ and volume by $8$ .

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