Similar Polygons and Scale Factor — Mathematics Knowledge System

Overview

Two polygons are similar when one is an enlargement or reduction of the other: corresponding angles are equal, and corresponding sides are all in the same ratio, called the scale factor. Similarity is a weaker condition than congruence — congruent figures are similar with scale factor 1, but similar figures need not be the same size, only the same shape. Because every linear measurement (side length, perimeter, diagonal) scales by the factor k, while area scales by k² and volume (for similar solids) scales by k³, scale factor is the single number that governs how all the other measurements of a shape change together.

Intuition

Think of a photocopier's zoom setting: enlarging a page by 150% multiplies every length on the page by 1.5, but it does not change any angle — the picture just gets bigger while keeping its exact shape. That zoom percentage is the scale factor. Because area is length times length, doubling every length (scale factor 2) quadruples the area (2² = 4), not just doubles it — this is why a poster twice as tall and twice as wide as a photo needs four times the paper, not two times.

Interactive Graph

Formal Definition

Definition

For two similar polygons with scale factor k (ratio of a length in the image to the corresponding length in the original), corresponding angles α_i, perimeters P, and areas A:

\alpha_i^{\text{image}} = \alpha_i^{\text{original}}

Corresponding angles are equal

\frac{\text{side}_i^{\text{image}}}{\text{side}_i^{\text{original}}} = k \quad \text{for every corresponding pair}

Corresponding sides share one common ratio k

P_{\text{image}} = k \cdot P_{\text{original}}

Perimeter scales linearly with k

A_{\text{image}} = k^2 \cdot A_{\text{original}}

Area scales with the square of k

Notation

Notation	Meaning
$\sim$	Is similar to (e.g. △ABC ~ △DEF)
$k$	Scale factor: ratio of a length in one figure to the corresponding length in the other

Properties

AA (Angle-Angle) similarity criterion for triangles

\text{If two angles of one triangle equal two angles of another, the triangles are similar.}

Condition: Sufficient for triangles because the third angle is then forced to match (angle-sum theorem).

SSS and SAS similarity criteria

\text{Triangles are similar if all three side ratios are equal (SSS), or if two side ratios are equal and the included angle matches (SAS).}

Perimeter and area scaling

P_{\text{image}} = kP_{\text{original}}, \qquad A_{\text{image}} = k^2 A_{\text{original}}

Condition: k is the ratio of any pair of corresponding linear measurements.

Example: Scale factor 3 triples the perimeter but multiplies area by 3² = 9.

Applications

Scale models and blueprints use a fixed scale factor so every dimension can be recovered by a single multiplication, while angles (and therefore the design's proportions) stay identical to the real structure.

Worked Examples

The shortest sides correspond (4 in ABC to 10 in DEF), giving the scale factor.
$k = \frac{10}{4} = 2.5$
Multiply the other two sides of ABC by k.
$6 \times 2.5 = 15, \qquad 8 \times 2.5 = 20$

Answer: Scale factor k = 2.5; the other sides of DEF are 15 and 20.

Practice Problems

Difficulty 3/10

Two similar pentagons have corresponding sides 5 and 15. If the smaller pentagon has perimeter 22, find the perimeter of the larger pentagon.

Difficulty 4/10

Two similar triangles have areas 9 and 81. Find the scale factor from the smaller triangle to the larger one.

Difficulty 5/10

A model of a building is built at a scale factor of 1/50 (every length on the model is 1/50 of the real length). The model's footprint has area 40 square inches. What is the real building's footprint area, in square inches?

Common Mistakes

Common Mistake

Assuming area scales the same way perimeter does (both by k).

Perimeter (a 1-dimensional sum of lengths) scales by k, but area (a 2-dimensional measure) scales by k² — always square the scale factor for area comparisons.

Common Mistake

Matching up sides or angles that are not actually corresponding, just because the shapes look similar.

Correspondence must respect the vertex order stated in the similarity (e.g. △ABC ~ △DEF means A↔D, B↔E, C↔F); mismatching the pairs gives a wrong scale factor even if the shapes are truly similar.

Quiz

Summary

Similar polygons have equal corresponding angles and proportional corresponding sides, with the constant ratio called the scale factor k.
Perimeter scales linearly with k; area scales with k².
For triangles, AA, SSS, and SAS are sufficient criteria to establish similarity.
Congruent figures are the special case of similar figures with scale factor exactly 1.
Scale factor and its square (for area) are the tools behind scale models, blueprints, and image resizing.

References

WebsiteWikipedia — Similarity (geometry)