01 / 21

Similarity &
Scale Problems

Learn to use proportional reasoning and scale factors to solve real-world problems with maps, models, and blueprints.

Pre-Algebra

What Does "Similar" Mean?

Two figures are similar when they have the same shape but not necessarily the same size. Every angle stays the same, and every side is multiplied by the same number: the scale factor.

A B C
\(6 \text{ cm}\)
\(5 \text{ cm}\)
\(5 \text{ cm}\)
 D E F
\(12 \text{ cm}\)
\(10 \text{ cm}\)
\(10 \text{ cm}\)
\(\times 2\)

The Scale Factor

A scale factor is the ratio that tells you how many times larger (or smaller) one figure is compared to another. If the scale factor from Figure A to Figure B is \(k\), then:

\(k > 1\)
The copy is larger than the original: an enlargement.
\(k = 1\)
The figures are congruent: same shape and same size.
\(k < 1\)
The copy is smaller than the original: a reduction.

Where Do We Use Scale?

Scale factors appear every time we need to represent something too large or too small to work with at full size.

Maps
"1 cm = 50 km" shrinks a country to fit on paper.
Blueprints
"1 in = 4 ft" lets an architect draw an entire building.
Models
A 1 : 200 airplane model is 200 times smaller than the real plane.

The Proportion Rule

When two figures are similar, the ratios of corresponding sides are equal. We write this as a proportion:

Small
\(5 \text{ in}\)
\(3 \text{ in}\)
 Large
\(11 \text{ in}\)
\(x\)
 scale
\(\frac{5}{11} = \frac{3}{x} \implies x = 6.6 \text{ in}\)

Cross Multiply to Solve

Once you set up a proportion, use cross multiplication to find the unknown value.

3 5 = x 11
\(3 \times 11 = 5 \times x\)
\(33 = 5x\)
\(x = 6.6\)

Similar vs Congruent

Similar (\(\sim\)) Congruent (\(\cong\))
Shape Same Same
Size Can differ Must match
Angles All equal All equal
Sides Proportional Equal
Scale factor Any \(k > 0\) \(k = 1\)

How to Solve Scale Problems

1
Identify the scale. Write it as a ratio: model : real (e.g., 1 cm : 50 km).
2
Match the units. Make sure both sides of the ratio use the same units, or convert.
3
Set up a proportion. Put known values on one side, the unknown on the other.
4
Cross multiply and solve. Multiply diagonally, then divide to isolate the variable.
5
Label your answer with units. Always state the unit (cm, km, ft, etc.).

Example 1: Map Distance

A map uses the scale 1 cm = 25 km. Two cities are 7.4 cm apart on the map. What is the real distance?

MAP City A City B
\(7.4 \text{ cm}\)
\(1 \text{ cm} = 25 \text{ km}\)

Example 2: Model Building

An architect builds a model of a skyscraper at a scale of 1 : 200. The real building is 90 m tall. How tall is the model?

Real
\(90 \text{ m}\)
 Model
\(? \text{ cm}\)
 1 : 200

Example 3: Blueprint Room

On a blueprint, 1 inch = 4 feet. A bedroom measures \(3.5 \text{ in}\) \(\times\) \(2.75 \text{ in}\) on the blueprint. Find the real dimensions in feet.

1
Length: \(3.5 \times 4 = \) 14 ft
2
Width: \(2.75 \times 4 = \) 11 ft
3
Real room: \(14 \text{ ft} \times 11 \text{ ft}\)

Handling Different Units

Sometimes the scale mixes units: "1 cm = 50 km." Before cross-multiplying, decide which unit you want your answer in, and keep your proportion consistent.

Option A
Keep the mixed units and solve directly: \(\frac{1\text{ cm}}{50\text{ km}} = \frac{x\text{ cm}}{d\text{ km}}\).
Option B
Convert to the same unit first (e.g., 50 km = 5,000,000 cm), then the scale factor is a pure number.

Finding the Scale Factor

Sometimes you are given two corresponding measurements and asked to find the scale. Divide one by the other, making sure units match first.

\(6 \text{ cm}\)
\(15 \text{ cm}\)
\(k = ?\)
\(k = \frac{15}{6}\)
\(k = 2.5\)

Area Changes by \(k^2\)

When lengths scale by \(k\), areas scale by \(k^2\). This is because area depends on two dimensions.

\(2 \text{ cm}\)
\(2\)
\(A = 4 \text{ cm}^2\)
\(\times 3\)
\(6 \text{ cm}\)
\(6\)
\(A = 36 \text{ cm}^2 = 4 \times 9\)

Common Mistakes

✕ Mismatched Units
Mixing cm and km in the same ratio without converting leads to answers that are off by factors of 1,000 or more.
✕ Flipped Ratio
Writing model/real on one side but real/model on the other. Both fractions must be set up the same way.
✕ Using \(k\) for Area
Multiplying an area by \(k\) instead of \(k^2\). Lengths scale by \(k\), but areas scale by \(k^2\).

Scale Problem Cheat Sheet

Scenario Set Up Solve
Map to Real \(\frac{1\text{ cm}}{k\text{ km}} = \frac{d_{\text{map}}}{x}\) Cross multiply
Model to Real \(\frac{1}{n} = \frac{\text{model}}{x}\) \(x = \text{model} \times n\)
Find Scale \(k = \frac{\text{larger}}{\text{smaller}}\) Divide (same units)
Scale Area \(A_{\text{big}} = k^2 \times A_{\text{small}}\) Square the factor

Question 1 of 5

A map has a scale of 1 cm = 40 km. Two towns are 6.5 cm apart on the map. What is the real distance?

Question 2 of 5

A model car is built at a scale of 1 : 24. The real car is 4.8 m long. How long is the model?

Question 3 of 5

Two similar rectangles have widths of 5 cm and 15 cm. If the smaller rectangle's length is 8 cm, what is the larger rectangle's length?

Question 4 of 5

A blueprint uses the scale 1 in = 3 ft. A hallway measures 4.5 in long on the blueprint. What is its real length?

Question 5 of 5

If two similar triangles have a scale factor of 4, how many times larger is the area of the bigger triangle?