Scale Factors and Similar Figures

A scale factor is the ratio that tells you how much larger or smaller one object is compared to another. Scale factors power everything from model airplanes to architectural blueprints to GPS navigation. If you can work with proportions, you are already equipped to handle scale factors — this topic builds directly on that foundation.

What Is a Scale Factor?

A scale factor is the constant ratio between corresponding measurements of two similar objects. If every dimension of a drawing is exactly 10 times smaller than the real object, the scale factor is $\frac{1}{10}$.

$\text{Scale Factor} = \frac{\text{New (or model) dimension}}{\text{Original (or real) dimension}}$

• A scale factor greater than 1 means the new figure is larger (an enlargement).
• A scale factor less than 1 means the new figure is smaller (a reduction).
• A scale factor equal to 1 means both figures are the same size.

Example 1: Finding the Scale Factor

A photograph is 4 inches wide. The original poster is 20 inches wide. What is the scale factor from the poster to the photograph?

$\text{Scale Factor} = \frac{\text{Photo width}}{\text{Poster width}} = \frac{4}{20} = \frac{1}{5}$

Answer: The scale factor is $\frac{1}{5}$ (the photo is one-fifth the size of the poster).

Example 2: Enlargement

A cell under a microscope is 0.05 mm wide. The image on screen is 15 mm wide. What is the scale factor?

$\text{Scale Factor} = \frac{15}{0.05} = 300$

Answer: The scale factor is $300$ — the image is 300 times larger than the actual cell.

Using Scale Factors to Find Missing Dimensions

When you know the scale factor and one dimension, you can find any other dimension by multiplying or dividing.

To go from original to scaled: Multiply by the scale factor.

$\text{Scaled dimension} = \text{Original dimension} \times \text{Scale Factor}$

To go from scaled to original: Divide by the scale factor.

$\text{Original dimension} = \frac{\text{Scaled dimension}}{\text{Scale Factor}}$

Example 3: Finding a Model Dimension

A model car is built at a $1 : 24$ scale. The real car is 192 inches long. How long is the model?

$\text{Model length} = 192 \times \frac{1}{24} = 8 \text{ inches}$

Answer: The model car is 8 inches long.

Example 4: Finding the Real Dimension

On an architect’s drawing, a wall is 3.5 inches long. The drawing uses a scale of $\frac{1}{4}$ inch = 1 foot. What is the actual wall length?

Each $\frac{1}{4}$ inch on the drawing equals 1 foot in reality.

$\text{Actual length} = 3.5 \div \frac{1}{4} = 3.5 \times 4 = 14 \text{ feet}$

Answer: The actual wall is 14 feet long.

Map Scales

Maps use scale factors to represent large distances in a small space. A map scale might say:

• 1 inch = 50 miles (a verbal scale)
• 1 : 250,000 (a representative fraction — 1 unit on the map equals 250,000 of the same units in reality)

Example 5: Using a Map Scale

A map has a scale of 1 inch = 30 miles. Two towns are 4.5 inches apart on the map. What is the actual distance?

$\text{Actual distance} = 4.5 \times 30 = 135 \text{ miles}$

Answer: The towns are 135 miles apart.

Example 6: Working Backward from a Map

On a map where 1 inch = 20 miles, you need to mark two cities that are 90 miles apart. How far apart should they be on the map?

$\text{Map distance} = \frac{90}{20} = 4.5 \text{ inches}$

Answer: The cities should be 4.5 inches apart on the map.

Blueprint Scales

Blueprints for buildings, electrical systems, and mechanical parts use the same concept. Common blueprint scales include:

Example 7: Blueprint Calculation

An electrician is reading a blueprint where $\frac{1}{4}$” = 1’. A conduit run measures 6.5 inches on the blueprint. What is the actual length?

Step 1: Determine how many quarter-inches fit in 6.5 inches.

$6.5 \div 0.25 = 26$

Step 2: Each quarter-inch represents 1 foot, so the actual length is 26 feet.

Or equivalently: $6.5 \times 4 = 26$ feet.

Answer: The conduit run is 26 feet in actual length.

Similar Figures

Two figures are similar if they have the same shape but possibly different sizes. This means:

1. All corresponding angles are equal.
2. All corresponding sides are in the same ratio (the scale factor).

Example 8: Finding a Missing Side in Similar Triangles

Triangle ABC is similar to Triangle DEF. In Triangle ABC, the sides are 6, 8, and 10. In Triangle DEF, the shortest side is 9. Find the other two sides.

Step 1: Identify the scale factor. The shortest side of ABC is 6, and the shortest side of DEF is 9.

$\text{Scale Factor} = \frac{9}{6} = \frac{3}{2} = 1.5$

Step 2: Multiply each side of ABC by 1.5.

$8 \times 1.5 = 12 \qquad 10 \times 1.5 = 15$

Answer: The sides of Triangle DEF are 9, 12, and 15.

Verification: $\frac{9}{6} = \frac{12}{8} = \frac{15}{10} = 1.5$. All ratios are equal. Checks out.

Real-World Application: Carpentry — Building from Plans

A carpenter is building a deck from a plan drawn at $\frac{1}{2}$” = 1’ scale. On the plan, the deck measures 7 inches by 5 inches. What are the actual dimensions?

$\text{Length} = 7 \times 2 = 14 \text{ feet}$

$\text{Width} = 5 \times 2 = 10 \text{ feet}$

Answer: The actual deck is 14 feet by 10 feet.

The carpenter can also calculate the actual area:

$\text{Area} = 14 \times 10 = 140 \text{ square feet}$

Important note about area: When linear dimensions are multiplied by a scale factor $k$, the area is multiplied by $k^2$. On the plan, the area is $7 \times 5 = 35$ square inches. The scale factor for length is 2 (feet per inch), but the area scale factor is $2^2 = 4$… well, the units also change, so it is best to compute actual dimensions first, then find area.

Real-World Application: Electrician — Panel Layout

An electrician reviews a blueprint ($\frac{1}{4}$” = 1’) showing a 100-foot wire run from a panel to an outlet. On the blueprint, this run should measure:

$\frac{100}{4} = 25 \text{ inches}$

The electrician can verify measurements on the blueprint to ensure the wire run is drawn correctly before ordering materials.

Common Mistakes to Avoid

1. Confusing which direction the scale goes. A $1 : 50$ scale means 1 unit on the model equals 50 units in reality. To find the real size, multiply by 50. To find the model size, divide by 50.

2. Forgetting to use the same units. If a blueprint says $\frac{1}{4}$” = 1’, your blueprint measurement must be in inches and your answer will be in feet. Do not mix the two.

3. Applying the linear scale factor to area. If a scale factor is 3, the area ratio is $3^2 = 9$, not 3. A figure that is 3 times as wide and 3 times as tall has 9 times the area.

4. Misidentifying corresponding sides. In similar figures, the smallest side corresponds to the smallest side, the largest to the largest. Do not match sides randomly.

Practice Problems

Key Takeaways

• A scale factor is the ratio of a model/drawing dimension to the real dimension (or vice versa).
• Scale factors greater than 1 are enlargements; less than 1 are reductions.
• To find a missing dimension, multiply (or divide) by the scale factor.
• Map and blueprint scales are practical applications of scale factors used daily in trades and navigation.
• Similar figures have equal angles and proportional sides — the scale factor applies to every corresponding pair of sides.
• Area scales by the square of the linear scale factor.

Return to Pre-Algebra for more topics in this section.