Similar Polygons

Definition

Two polygons are similar ($\sim$) if and only if:

1. All pairs of corresponding angles are congruent.
2. All pairs of corresponding sides are proportional.

• How to read: “Polygon A is similar to polygon B.”
• Meaning: Same shape, possibly different size—angles match and sides scale by a constant ratio.

Why It Matters

Similarity is the ‘scaling law’ of geometry; it allows us to understand the properties of a system by studying its smaller versions, which is the basis for architectural models and proportional engineering.

Core Concepts

• Similarity vs. Congruence: Similar figures have the same shape but not necessarily the same size. Congruence is a special case of similarity where the ratio of sides is $1:1$.

  • How to read: “Ratio one to one.”
  • Meaning: Congruent figures are similar with scale factor $k = 1$.

• Constant of Proportionality ($k$): The fixed ratio of the lengths of corresponding sides.

  • $k > 1$: Enlargement (Stretch).
  • $0 < k < 1$: Reduction (Shrink).
  • How to read: “K greater than one is enlargement; zero less than k less than one is reduction.”
  • Meaning / when to use: Every corresponding side in the image is $k$ times the original.

• Scale Factor: Often used interchangeably with $k$.

• Perimeters of Similar Polygons: The ratio of the perimeters of two similar polygons is equal to the ratio of any two corresponding sides.

• Areas of Similar Polygons: The ratio of the areas of two similar polygons is equal to the square of the ratio of any two corresponding linear segments (sides, altitudes, medians, perimeters). $\frac{Area_1}{Area_2} = \left(\frac{s_1}{s_2}\right)^2$

  • How to read: “The ratio of area one to area two equals the square of the quantity s one divided by s two.”
  • Meaning / when to use: Areas scale as the square of the linear scale factor—double the sides, quadruple the area.

• Naming Convention: When naming similar polygons (e.g., $ABCD \sim EFGH$), the order of vertices must correspond to the matching angles and sides.

  • How to read: “A-B-C-D is similar to E-F-G-H.”
  • Meaning: Vertex order matters—$A$ corresponds to $E$, $B$ to $F$, etc.

Connected Concepts

• Ratio
• Proportion
• Similar Triangles
• Quadrilateral Hierarchy