Scale Factor: Perimeter, Area & Volume

Q: Does surface area scale like area or volume?

Like area — by $k^2$. Surface area is still a two-dimensional measure even though it wraps a 3D solid.

When you enlarge or shrink a figure by a scale factor $k$ , not everything grows at the same rate. Lengths scale by $k$ , but areas scale by $k^2$ and volumes by $k^3$ . This surprises almost everyone the first time.

Double a shape's dimensions and its perimeter doubles, but its area quadruples and its volume grows eightfold. Knowing which power to use saves you from a very common wrong answer.

The three powers

Scale a figure by $k$ . Any length — perimeter, side, radius — is multiplied by $k$ . Any area — surface area included — is multiplied by $k^2$ . Any volume is multiplied by $k^3$ .

The pattern follows the dimension: length is 1D so $k^1$ , area is 2D so $k^2$ , volume is 3D so $k^3$ . Match the power to how many dimensions the quantity measures.

The figure shows why area uses $k^2$ : doubling a square's side ( $k = 2$ ) fits four copies of the original square inside, so the area is multiplied by $2^2 = 4$ .

Using it both directions

To go from original to image, multiply by the right power of $k$ . To go backwards — image to original — divide by that power.

You can also find $k$ from a ratio: if two similar solids have volumes in the ratio $8 : 1$ , the scale factor is $\sqrt[3]{8} = 2$ , and their surface areas are then in the ratio $2^2 : 1 = 4 : 1$ .

Worked examples

Example 1: area under a dilation

A triangle has area $6$ and is dilated by scale factor $3$ . Find the new area.

Area scales by k squared

k^2 = 3^2 = 9

Multiply the original area

6 \cdot 9

Simplify

54

Answer: $54$ square units

Example 2: volume from a scale factor

A solid has volume $10$ and is scaled by factor $2$ . Find the new volume.

Volume scales by k cubed

k^3 = 2^3 = 8

Multiply the original volume

10 \cdot 8

Simplify

80

Answer: $80$ cubic units

Example 3: scale factor from a volume ratio

Two similar solids have volumes in the ratio $125 : 8$ . Find the ratio of their surface areas.

Volume ratio is k cubed, so take the cube root

k = \sqrt[3]{\dfrac{125}{8}} = \dfrac{5}{2}

Surface area scales by k squared

k^2 = \left(\dfrac{5}{2}\right)^2 = \dfrac{25}{4}

Write it as a ratio

25 : 4

Answer: $25 : 4$

Try one yourself

Common questions

Why does area scale by k squared, not k?

Area is length times length. Each length is multiplied by $k$ , so the product is multiplied by $k \cdot k = k^2$ .

How do I find the scale factor from a volume ratio?

Take the cube root of the volume ratio. If volumes are $27 : 1$ , then $k = \sqrt[3]{27} = 3$ .

Does surface area scale like area or volume?

Like area — by $k^2$ . Surface area is still a two-dimensional measure even though it wraps a 3D solid.

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Scale Factor: Perimeter, Area & Volume

The three powers

Using it both directions

Worked examples

Example 1: area under a dilation

Example 2: volume from a scale factor

Example 3: scale factor from a volume ratio

Try one yourself

Common questions

Keep learning

Surface Area & Volume of Spheres

Dilations & Scale Factor

Modeling with Geometry: Density & Design