Scale factor effect on length (k), area (k²), volume (k³)
Scale Factor and Length, Area, and Volume
Scale Factor and Length, Area, and Volume is a topic in Similarity in the Common Core State Standards. It is aligned to Standard G-SRT.1, which requires students to understand that similarity transformations scale length by k, so area scales by k squared and volume by k cubed.
Scaling a figure by a factor \(k\) multiplies its lengths by \(k\), its areas by \(k^2\), and its volumes by \(k^3\).
Theory
When a figure is scaled by a factor \(k\), the effect grows with dimension:
- Length scales by \(k\).
- Area scales by \(k^2\).
- Volume scales by \(k^3\).
So doubling every length (\(k=2\)) multiplies area by \(4\) and volume by \(8\).
The scaling rules:
How to use the scaling rules
- Forward: square \(k\) for area, cube \(k\) for volume.
- Backward: take the square root of an area ratio or cube root of a volume ratio to get \(k\).
- Apply the ratio to the known measurement.
Area scales by \(k^2\).
| \(k^2\) | \(=\) | \(3^2=9\) |
The larger area is \(9\) times the smaller.
Volume scales by \(k^3\).
| \(k^3\) | \(=\) | \(2^3=8\) |
The larger volume is \(8\) times the smaller.
The scale factor is the square root of the area ratio.
| \(k\) | \(=\) | \(\sqrt{\dfrac{25}{4}}=\dfrac{5}{2}\) |
The scale factor is the cube root of the volume ratio.
| \(k\) | \(=\) | \(\sqrt[3]{27}=3\) |
Common pitfalls
Frequently asked questions
How does area change with the scale factor?
Area scales by \(k^2\): if lengths triple, area becomes \(9\) times as large.
How does volume change with the scale factor?
Volume scales by \(k^3\): if lengths double, volume becomes \(8\) times as large.
How do you find the scale factor from an area ratio?
Take the square root of the area ratio.
How do you find the scale factor from a volume ratio?
Take the cube root of the volume ratio.