Similarity Ratio (Scale Factor)

Table of Contents

• What is the similarity ratio
• How to calculate the similarity ratio
• What the value of k tells us
• Effect on area and volume
• Examples
• Related Articles

---

What is the similarity ratio

If two triangles $ABC$ and $A^{\prime }{B}^{\prime }{C}^{\prime }$ are similar ($△ABC\sim △A^{\prime }{B}^{\prime }{C}^{\prime }$), then there exists a positive number $k$ such that:

This number $k$ is called the similarity ratio (or scale factor). It expresses how many times the sides of one triangle are larger (or smaller) than the sides of the other.

💡 Remember: The similarity ratio is always a positive number ($k>0$).

---

How to calculate the similarity ratio

We find the similarity ratio by dividing corresponding sides:

Step by step:

• Determine which sides correspond to each other (opposite equal angles)
• Divide any pair of corresponding sides
• Verify that the ratio is the same for all three pairs

Example: $△ABC$: $a=4$, $b=6$, $c=8$ and $△A^{\prime }{B}^{\prime }{C}^{\prime }$: $a^{\prime }=6$, $b^{\prime }=9$, $c^{\prime }=12$

Verification: $\frac{b^{\prime }}{b}=\frac{9}{6}=1.5$ and $\frac{c^{\prime }}{c}=\frac{12}{8}=1.5$

---

What the value of k tells us

• $k>1$ — the second triangle is larger than the first (enlargement)
• $k<1$ — the second triangle is smaller than the first (reduction)
• $k=1$ — the triangles are congruent (the same size) — similarity is a special case of congruence

---

Effect on area and volume

The scale factor also affects the area and volume of figures:

• The ratio of areas of similar figures is $k^2$
• The ratio of volumes of similar solids is $k^3$

Example: If $k=2$, then:

• Sides are $2\times$ larger
• Area is $2^2=4\times$ larger
• Volume is $2^3=8\times$ larger

⚠️ Caution: A common mistake is confusing the ratio of sides with the ratio of areas. If the sides are $2\times$ larger, the area is not $2\times$ but $4\times$ larger!

---

Examples

Example 1: Triangle $ABC$ has sides $a=5$ cm, $b=7$ cm, $c=9$ cm. Triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ has sides $a^{\prime }=15$ cm, $b^{\prime }=21$ cm, $c^{\prime }=27$ cm. Find $k$.

Triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ is $3\times$ larger.

Example 2: Triangles $ABC\sim A^{\prime }{B}^{\prime }{C}^{\prime }$, $a=12$ cm, $a^{\prime }=8$ cm. What is $k$?

Triangle $A^{\prime }{B}^{\prime }{C}^{\prime }$ is smaller.

Example 3: Triangles $ABC\sim A^{\prime }{B}^{\prime }{C}^{\prime }$, $k=2.5$, $b=6$ cm. What is $b^{\prime }$?

👉 More examples: Triangle Similarity – Solved Examples

---

Related Articles

• Comprehensive guide: Triangle Similarity – Comprehensive Guide
• Similarity of figures: Similarity of Geometric Figures
• Similarity theorems: Triangle Similarity Theorems – SSS, SAS, AA
• Solved examples: Triangle Similarity – Solved Examples

---

Practice

• 🔢 Similarity ratio: Calculate the Similarity Ratio
• 📐 Missing sides: Find the Missing Side
• 🔍 Similarity theorems: Identify the Similarity Theorem
• 🧮 Mixed problems: Mixed Similarity Problems