Perimeter & Area Scaling

Statement

When a 2D shape is scaled by a scale factor \(k\):

• The perimeter is multiplied by \(k\): \(\; P' = kP\).
• The area is multiplied by \(k^2\): \(\; A' = k^2A\).

Why it’s true

• All side lengths scale directly with the scale factor \(k\), so the perimeter (sum of sides) scales by \(k\).
• The area is proportional to the square of lengths (since area involves two dimensions), so the area scales by \(k^2\).

Recipe (how to use it)

1. Identify the scale factor \(k\).
2. To find new perimeter: multiply old perimeter by \(k\).
3. To find new area: multiply old area by \(k^2\).

Spotting it

This appears in enlargement problems, similarity questions, and exam questions about ratios of areas and perimeters.

Common pairings

• Similar triangles, rectangles, or polygons.
• Circle scaling (perimeter = circumference, area = πr²).
• Maps, models, and real-life enlargements or reductions.

Mini examples

1. Given: A square has perimeter 40. Enlarged by scale factor 3. Answer: New perimeter = \(3 \times 40 = 120\).
2. Given: Rectangle has area 20. Enlarged by scale factor 4. Answer: New area = \(4^2 \times 20 = 320\).

Pitfalls

• Forgetting to square the scale factor for area.
• Confusing perimeter and area scaling rules.
• Mixing up ratios of sides with ratios of areas.

Exam strategy

• Always check whether the question is about perimeter or area.
• Write down \(P' = kP\) and \(A' = k^2A\) explicitly to avoid mistakes.
• If a question gives you ratios of areas, take the square root to find the scale factor.

Summary

Under enlargement by a scale factor \(k\), perimeters scale by \(k\) and areas scale by \(k^2\). This is essential in similarity and enlargement problems.