Ratios of Perimeters & Areas
When two figures are similar with scale factor , their perimeters have ratio and their areas have ratio . This “squaring rule” for area is one of the most powerful consequences of similarity.
Introduction
If you double every side of a rectangle, the perimeter doubles — but the area quadruples. This pattern holds for all similar figures: perimeter scales linearly with , but area scales with . Understanding this distinction is critical for real-world applications like painting, tiling, and material costs.
Past Knowledge
Dilations & scale factor (7.1.3). Similarity (7.2). Area formulas (Unit 10 preview).
Today's Goal
Compute perimeter and area ratios from the scale factor, and vice versa.
Future Success
Surface area & volume ratios (3D similarity), optimization problems.
Key Concepts
Perimeter Ratio
If two similar figures have a scale factor of , then:
Perimeter scales the same way as individual sides — by the first power of .
Area Ratio
For the same similar figures:
Area involves two dimensions, so the scale factor gets squared.
Working Backwards
Given the area ratio , the scale factor is . Given the perimeter ratio, the scale factor is the ratio directly.
Theorem & Proof
Two-Column Proof: Area Ratio of Similar Figures
Given: with scale factor
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | Definition of similarity with scale factor | |
| 2 | Let be the height of from to , and be the corresponding height of | Corresponding altitudes exist in both triangles |
| 3 | Corresponding altitudes of similar triangles have the same ratio as corresponding sides | |
| 4 | and | Area formula: |
| 5 | Substituting and from steps 1, 3 |
∎ Area scales as the square of the linear scale factor. Both the base and height scale by , giving .
Worked Examples
Scale Factor → Perimeter & Area
Two similar pentagons have a scale factor of . The smaller has perimeter 20 and area 30. Find the larger's perimeter and area.
Perimeter:
Area:
Perimeter = 60, Area = 270
Area Ratio → Scale Factor
Two similar triangles have areas 50 and 200. Find the scale factor and the ratio of their perimeters.
Area ratio:
Scale factor:
Perimeter ratio: (same as scale factor)
, perimeter ratio =
Cost Problem
A small pizza (10" diameter) costs $8. A large pizza (16" diameter) is similar in shape. If cost is proportional to area, what should the large cost?
Scale factor:
Area ratio:
Cost:
$20.48 — the large has 2.56× the area of the small!
Common Pitfalls
Using k for Area Instead of k²
This is the #1 mistake. If , the area is larger, not . Always square the scale factor for area.
Using k² for Perimeter
Perimeter is a length (1D), so it scales by , not . Only area (2D) uses the square.
Real-Life Applications
Painting a House
A model house is 1/20 scale. The model surface needs 0.5 square feet of paint. The real house needs square feet of paint — that's the area ratio in action!
Pizza Economics
A 16-inch pizza has times the area of a 12-inch pizza. If it costs less than 1.78× the price, the larger pizza is a better deal per square inch.
Practice Quiz
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