How to Find Similarity and Ratios? (+FREE Worksheet!)

How to find similarity and ratios, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

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Scale Drawings and Similarity Ratios: what to notice and how to work it

Proportional skill

Ratio and proportion problems compare quantities. The key is keeping units aligned so the same kind of quantity sits across from the same kind of quantity.

What to notice first

Find the unit rate or common multiplier first. Once one unit is clear, the rest of the proportion follows naturally.

Common student mistake

Do not cross-multiply before checking the order of the ratios. Mixed-up units can produce a neat but wrong equation.

Key formulas and cues

$\frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc$

$\text{unit rate}=\frac{\text{amount}}{\text{1 unit}}$

$\text{percent}=\frac{\text{part}}{\text{whole}}\cdot100\%$

A reliable path

Label unitsWrite what each number measures.
Build matching ratiosPlace the same units in the same positions.
Solve and interpretUse cross-products or a unit rate, then attach the correct unit.

Worked examples

Find a unit rate

Example: 3 notebooks cost $12

Divide total cost by number of notebooks.
12 divided by 3 is 4.
Attach the unit.

Answer: $4 per notebook

Solve a proportion

Example: $\frac{5}{8}=\frac{x}{24}$

The second denominator is 3 times 8.
Multiply 5 by 3.
Keep the ratios matched.

Answer: $x=15$

Open pop-up practice

Try one before moving on

Try: 5 tickets cost $40. What is the cost per ticket?

Answer: $8 per ticket.

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

When two figures are similar, their angles are equal and their corresponding side lengths form equal ratios called a scale factor. Understanding similarity and ratios lets you find missing side lengths, calculate scale distances on maps, and solve many real-world geometry problems in Algebra 1 and beyond.

What Is Similarity?

Two figures are similar (~) if they have the same shape but not necessarily the same size. Similar figures have:

All corresponding angles equal.
All corresponding side lengths in the same ratio (the scale factor).

For example, if triangle ABC ~ triangle DEF, then $\color{blue}{\frac{\text{ AB }}{\text{ DE }} = \frac{\text{ BC }}{\text{ EF }} = \frac{\text{ AC }}{\text{ DF }}}$.

Key Concepts: Scale Factor and Ratios

Scale Factor

The scale factor is the ratio of any pair of corresponding side lengths. If triangle 1 has sides 4, 6, and 8 and triangle 2 has sides 12, 18, and 24, the scale factor is $\color{blue}{\frac{4}{12} = \frac{1}{3}}$ (or 3 if going the other direction).

Setting Up the Proportion

To find a missing side, set corresponding sides equal as a proportion and solve with cross-multiplication.

If $\color{blue}{\frac{5}{10} = \frac{x}{14}}$, cross-multiply: $\color{blue}{10x = 70}$, so $\color{blue}{x = 7}$.
If $\color{blue}{\frac{8}{12} = \frac{6}{x}}$, cross-multiply: $\color{blue}{8x = 72}$, so $\color{blue}{x = 9}$.

Simplifying the Ratio

Always simplify side ratios to confirm similarity. A ratio of 6:9 simplifies to $\color{blue}{2:3}$; a ratio of 8:12 also simplifies to $\color{blue}{2:3}$, confirming the two pairs of sides are proportional.

Step-by-Step Summary

Identify the corresponding sides of the two similar figures.
Write the ratio of a known pair of corresponding sides to find the scale factor.
Set up a proportion using the scale factor and the unknown side.
Cross-multiply and solve for the unknown.
Check by verifying all ratios of corresponding sides are equal.

Watch: Equivalent Ratios in Similar Shapes

Khan Academy shows how equivalent ratios arise from similar shapes in geometry:

Similarity and Ratios – Worked Examples

Example 1: Two similar rectangles have widths 4 cm and 12 cm. The length of the smaller rectangle is 6 cm. Find the length of the larger rectangle.

Scale factor: $\color{blue}{\frac{12}{4} = 3}$. Larger length = $\color{blue}{6 \times 3 = 18 \text{ cm }}$.

Example 2: Two similar triangles have sides in ratio 2:3. If the shorter triangle has a side of 8, find the corresponding side of the larger triangle.

Set up: $\color{blue}{\frac{2}{3} = \frac{8}{x}}$. Cross-multiply: $\color{blue}{2x = 24}$, so $\color{blue}{x = 12}$.

Example 3: Solve for x: $\color{blue}{\frac{5}{10} = \frac{x}{14}}$.

Cross-multiply: $\color{blue}{10x = 70}$, so $\color{blue}{x = 7}$.

Example 4: Solve for x: $\color{blue}{\frac{8}{12} = \frac{6}{x}}$.

Cross-multiply: $\color{blue}{8x = 72}$, so $\color{blue}{x = 9}$.

More Practice: Using Similarity to Estimate Ratios

Khan Academy demonstrates how to use similarity to estimate the ratio between side lengths:

Exercises for Similarity and Ratios

Find the missing side length in each pair of similar figures.

$\color{blue}{\frac{3}{6} = \frac{x}{10}}$
$\color{blue}{\frac{4}{6} = \frac{8}{x}}$
$\color{blue}{\frac{5}{15} = \frac{x}{9}}$
$\color{blue}{\frac{10}{15} = \frac{6}{x}}$
Two similar triangles have sides in ratio 3:5. The smaller has a side of 12. Find the corresponding larger side.
A scale model is built at a 1:20 ratio. If a wall on the model is 5 cm, how long is the real wall in cm?

Answers

$\color{blue}{x = 5}$
$\color{blue}{x = 12}$
$\color{blue}{x = 3}$
$\color{blue}{x = 9}$
$\color{blue}{20}$ (set up $\color{blue}{\frac{3}{5} = \frac{12}{x}}$; $\color{blue}{x = 20}$)
$\color{blue}{100 \text{ cm }}$ (set up $\color{blue}{\frac{1}{20} = \frac{5}{x}}$; $\color{blue}{x = 100}$)

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Frequently Asked Questions

What is a similarity ratio?

A similarity ratio is the ratio of matching side lengths in two similar figures, and it is also called the scale factor. If one triangle has sides 3, 4 and 5 and a similar triangle has sides 6, 8 and 10, every pair of matching sides compares as 1 to 2, so the similarity ratio is 1:2. The ratio stays the same for every pair of corresponding sides.

How do you find the similarity ratio?

Pick one pair of corresponding sides, write the first figure’s length over the second figure’s length, and simplify. For two similar rectangles with lengths 12 cm and 8 cm, the ratio is 12/8, which simplifies to 3/2, or 3:2. Check a second pair of sides, such as widths 6 cm and 4 cm, to confirm you get the same ratio before using it.

How do you find the similarity ratio in triangles?

First match the angles so you know which sides correspond, then divide a pair of matching sides. In similar triangles ABC and DEF, if AB = 5 and DE = 15, the ratio is 5/15 = 1/3. That means every side of triangle ABC is one third of the matching side in triangle DEF, so if BC = 7, then EF = 21.

How are perimeter and area ratios related in similar figures?

If two figures are similar with a scale factor of k, their perimeters compare in the same ratio k, but their areas compare in the ratio k². For example, two similar triangles with a side ratio of 2:3 have perimeters in the ratio 2:3 and areas in the ratio 4:9. Volumes of similar solids go one step further and compare as k³.

What is a similarity statement?

A similarity statement uses the symbol ~ to say two figures are similar, such as △ABC ~ △DEF. The order of the letters matters: A matches D, B matches E and C matches F. Once the statement is written, you can read off corresponding sides like AB and DE directly, which makes it much easier to set up the correct similarity ratio.

Is there a similarity and ratios worksheet?

Yes. The free worksheet for this lesson gives practice finding scale factors, missing side lengths and ratios in similar shapes. Try a few problems right after reading the worked examples, label which sides correspond in each pair of figures, and check your answers so you can spot whether a mistake came from matching the wrong sides or from the arithmetic.

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How to Find Similarity and Ratios? (+FREE Worksheet!)