Scale Factor: Definition, Formula & Examples
Scale Factor: Definition, Formula & Examples
TL;DR
The scale factor is the number you multiply every length of a figure by to get the matching length of a similar figure, equal to new length ÷ original length. This article covers the formula, scaling up versus down, dilation on the coordinate plane, the area (k²) and volume (k³) rules, and six worked examples.
What Is a Scale Factor?
A scale factor is the ratio by which every length of a figure is multiplied to make a similar figure. If a triangle with sides 3, 4, 5 becomes one with sides 6, 8, 10, every side was multiplied by 2, so the scale factor is 2. The two figures share the same shape — equal angles, proportional sides — but differ in size. This is precisely the relationship of similar figures, with the scale factor naming the ratio.
The Scale Factor Formula
The formula reads:
k = \frac{\text{dimension of the new figure}}{\text{dimension of the original figure}}.
Here k is the scale factor and the two dimensions are any pair of corresponding lengths. If k = 2, every length doubles; if k = \frac{1}{2}, every length halves; if k = 1, the new figure is the same size as the original (congruent, not just similar). Because the numerator and denominator carry the same units, k itself has no units. Order matters: the scale factor from A to B is the reciprocal of the scale factor from B to A.
How Do You Find the Scale Factor?
Divide a length on the new figure by the matching length on the original. If the figures are truly similar, every corresponding pair gives the same answer, so any one pair will do. To go the other way and find a missing length, multiply the original length by k.
Scaling Up vs Scaling Down
The value of k decides whether the figure grows, shrinks, or stays put.
| Scale factor | Effect | Where you see it |
|---|---|---|
| k > 1 | Enlargement (scaling up) | Photo enlargement, building from a model |
| k = 1 | No change (congruent figures) | Exact copies |
| 0 < k < 1 | Reduction (scaling down) | Architectural plans, map scales |
| k < 0 | Enlarge or reduce, plus a flip | Some coordinate dilations |
A scale factor of 3 makes the figure 3 times bigger; a scale factor of \frac{1}{3} makes it 3 times smaller. How far k sits from 1 measures how dramatic the change is.
Scale Factor in Dilations
On the coordinate plane, a dilation is the transformation that produces a scaled copy about a fixed centre C. Pick a centre and a scale factor k; each point P moves to a new point P' so that the vector from C is multiplied by k:
[ \vec{CP'} = k \cdot \vec{CP}. ]
In words: stretch or shrink every segment from the centre by the factor k, leaving the angles unchanged. When the centre is the origin, this is just ( P' = (kx, ky) ). A positive k keeps the image on the same side of the centre; a negative k sends it to the opposite side, flipping it through the centre as it scales.
The Area and Volume Rules
When a figure is scaled by a factor k, lengths, areas, and volumes do not all change by the same amount.
- Lengths multiply by k.
- Areas multiply by k².
- Volumes multiply by k³.
The reason is dimensional. An area is a product of two lengths, so scaling each length by k scales the area by k·k=k²; a volume is a product of three lengths, so it scales by k³. The general rule: a quantity built from n lengths scales by kⁿ. Double a square's sides and its area quadruples; double a cube's edges and its volume grows eightfold.
Examples of Scale Factor
Example 1 - A rectangle measures 5 cm by 8 cm and is enlarged by a scale factor of 3.
Multiply each length by 3: new width = 5 × 3 = 15 cm, new length = 8 × 3 = 24 cm.
Final answer: 15 cm by 24 cm.
Example 2 - A model car is built at a scale factor of \frac{1}{50} relative to the real car. The model's surface area is 400 cm². Find the real car's surface area.
A common first move is to multiply the model's area by 50 to undo the shrink: 400 × 50 = 20,000 cm².
Final answer: 1,000,000 cm² = 100 m².
Example 3 - A square has side length 7 cm and is enlarged by a scale factor of 4. Find the area of the new square.
The new side is 7 × 4 = 28 cm, so the new area is 28² = 784 cm².
Final answer: 784 cm².
Example 4 - On a map labelled "1 cm = 5 km", two towns are 8 cm apart. Find the real distance.
Final answer: 40 km.
Example 5 - A cone has volume 24 cm³. A larger, similar cone is built at a scale factor of 2. Find its volume.
Final answer: 192 cm³.
Example 6 - Triangle ABC has vertices A(0,0), B(4,0), C(0,6) and is dilated about the origin by a scale factor of -1.5. Find the image vertices and the area ratio.
Final answer: A'(0,0), B'(-6,0), C'(0,-9); area ratio 2.25.
Where the Scale Factor Shows Up
- Maps. A map marked "1:50,000" has a scale factor of \frac{1}{50,000}: 1 cm on paper is 500 m on the ground.
- Architectural and engineering drawings. Blueprints shrink buildings by factors like \frac{1}{100}.
- 3D printing and digital zoom. A "2× zoom" or a print scale is a literal scale factor applied to every dimension.
- Microscopy. A magnification of 400× is a scale factor of 400 from the real specimen to the viewed image.
Where Students Trip Up on Scale Factor
Mistake 1: Scaling area or volume by k instead of k² or k³
Where it slips in: Any "scale a figure by k, find the new area or volume" problem.
Don't do this: Multiply the area by k directly.
The correct way: Multiply area by k² and volume by k³.
Mistake 2: Inverting the scale-factor direction
Where it slips in: A model-and-real-object problem where it is unclear which way the factor runs.
Don't do this: Use the same factor for both directions.
The correct way: The factor from A to B is B's dimension over A's; the factor from B to A is its reciprocal.
Mistake 3: Reading a map scale as the scale factor without converting units
Where it slips in: A map labelled "1 cm = 1 km" read as a scale factor of "1 over 1".
Don't do this: Drop the units before dividing.
The correct way: Convert both sides to one unit first.
Key Takeaways
- The scale factor is new length ÷ original length, the multiplier that turns each original length into the matching new one.
- k > 1 enlarges, 0 < k < 1 reduces, k = 1 is congruence, and a negative k adds a flip through the centre.
- Area scales by k² and volume by k³ — the rule most often missed.