Scale Factor

Scale Factor

The ratio of any two corresponding lengths in two similar geometric figures. Note: The ratio of areas of two similar figures is the square of the scale factor. The ratio of volumes of two similar figures is the cube of the scale factor.

Key Formula

k = \frac{\text{length in new figure}}{\text{length in original figure}}

Where:

$k$ = The scale factor
$\text{length in new figure}$ = A side length (or any linear measurement) in the enlarged or reduced figure
$\text{length in original figure}$ = The corresponding side length in the original figure

Worked Example

Problem: Triangle ABC has sides of length 3 cm, 4 cm, and 5 cm. Triangle DEF is similar to triangle ABC and has a shortest side of 9 cm. Find the scale factor and the lengths of the other two sides of triangle DEF. Then find the ratio of their areas.

Step 1: Identify corresponding sides. The shortest side of ABC is 3 cm and the shortest side of DEF is 9 cm.

Step 2: Compute the scale factor from ABC to DEF by dividing the new length by the original length.

k = \frac{9}{3} = 3

Step 3: Multiply each side of ABC by the scale factor to find the remaining sides of DEF.

4 \times 3 = 12 \text{ cm}, \quad 5 \times 3 = 15 \text{ cm}

Step 4: The ratio of areas equals the square of the scale factor.

\frac{\text{Area of DEF}}{\text{Area of ABC}} = k^2 = 3^2 = 9

Answer: The scale factor is 3. The sides of triangle DEF are 9 cm, 12 cm, and 15 cm. The area of DEF is 9 times the area of ABC.

Another Example

Problem: A rectangular prism has dimensions 2 cm × 4 cm × 6 cm. A similar prism is created using a scale factor of 0.5. Find the new dimensions and the ratio of the volumes.

Step 1: Multiply each dimension by the scale factor of 0.5.

2 \times 0.5 = 1, \quad 4 \times 0.5 = 2, \quad 6 \times 0.5 = 3

Step 2: The new prism has dimensions 1 cm × 2 cm × 3 cm. Now find the ratio of volumes by cubing the scale factor.

\frac{V_{\text{new}}}{V_{\text{original}}} = k^3 = 0.5^3 = 0.125 = \frac{1}{8}

Step 3: Verify: the original volume is 48 cm³ and the new volume is 6 cm³.

\frac{6}{48} = \frac{1}{8} \checkmark

Answer: The new dimensions are 1 cm × 2 cm × 3 cm, and the new volume is one-eighth of the original volume.

Frequently Asked Questions

What does a scale factor less than 1 mean?

A scale factor between 0 and 1 means the new figure is smaller than the original — it is a reduction. For example, a scale factor of 0.5 means every length in the new figure is half the corresponding length in the original.

How do you find the scale factor between two similar figures?

Pick any pair of corresponding sides and divide the length in the new (or second) figure by the length in the original (or first) figure. This ratio is the scale factor. Make sure you match corresponding sides — the shortest with the shortest, the longest with the longest, and so on.

Scale Factor vs. Ratio

A ratio is any comparison of two quantities by division. A scale factor is a specific kind of ratio: it compares corresponding lengths of similar figures. Every scale factor is a ratio, but not every ratio is a scale factor. For instance, the ratio of a rectangle's length to its width is a ratio within one figure, not a scale factor between two figures.

Why It Matters

Scale factors are essential in real-world applications like reading maps, creating architectural blueprints, and building scale models. Knowing the scale factor lets you convert any measurement between the model and the real object. The squared and cubed relationships are especially important — they explain why doubling the dimensions of a box quadruples its surface area but increases its volume eightfold.

Common Mistakes

Mistake: Squaring the scale factor to find a missing length instead of using it directly.

Correction: The scale factor applies directly to lengths. You only square it for areas and cube it for volumes. If the scale factor is 3, a side that was 5 cm becomes 15 cm (not 45 cm).

Mistake: Reversing the order of the ratio and getting the reciprocal of the intended scale factor.

Correction: Always be clear about direction. The scale factor from figure A to figure B is (length in B) ÷ (length in A). If the scale factor from A to B is 3, then the scale factor from B to A is 1/3.

Related Terms

Ratio — General comparison that scale factor is a type of
Similar — Figures must be similar for scale factor to apply
Geometric Figure — The shapes being compared
Volume — Scales by the cube of the scale factor
Proportion — Equation stating two ratios are equal
Dilation — Transformation that enlarges or reduces by a scale factor
Area — Scales by the square of the scale factor