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Scale Factor
Effects

Discover how a scale factor transforms perimeter and area in scaled figures: one multiplies, the other squares.

Pre-Algebra

What Is a Scale Factor?

A scale factor is the ratio that describes how much larger or smaller a new figure is compared to the original. If every length in a figure is multiplied by a constant \( k \), we say the figure has been scaled by \( k \).

\( k > 1 \)
The figure gets larger: an enlargement.
\( k = 1 \)
The figure stays the same size: congruent.
\( 0 < k < 1 \)
The figure gets smaller: a reduction.

Seeing Scale in Action

Original
\( 3 \) cm
\( 3 \) cm
\( k = 2 \)
 Scaled
\( 6 \) cm
\( 6 \) cm
\( P: 12 \to 24 \)
\( (\times 2) \)
\( A: 9 \to 36 \)
\( (\times 4) \)

Why Does This Matter?

Scale factor effects show up everywhere in real life: architects shrink buildings to make blueprints, map makers reduce continents to fit on paper, and engineers enlarge tiny parts for manufacturing. Understanding how perimeter and area change helps you predict costs, material needs, and dimensions without measuring everything from scratch.

The Core Rules

Perimeter Rule
New Perimeter \( = k \times \) Original Perimeter

Perimeter scales by \( k \).
Area Rule
New Area \( = k^2 \times \) Original Area

Area scales by \( k^2 \).

Why \( k^2 \) for Area?

\( 1 \times 1 \)
\( A = 1 \)
\( \to \)
\( 3 \times 3 \)
\( A = 9 = 3^2 \times 1 \)
\( k = 3 \to \text{area} \times 9 \)
Each dimension scales by \( k \).
\( \text{Area} = \text{length} \times \text{width} \)
\( = (k \cdot l) \times (k \cdot w) \)
\( = k^2 \times (l \times w) \)

Perimeter vs. Area: Key Distinction

Perimeter Area
Dimension 1-D (linear) 2-D (squared)
Scale rule \( k \times P \) \( k^2 \times A \)
\( k = 3 \) \( \times 3 \) \( \times 9 \)
\( k = \frac{1}{2} \) \( \times \frac{1}{2} \) \( \times \frac{1}{4} \)
Units cm, m, in cm², m², in²

How to Apply the Rules

1
Identify the scale factor \( k \). This is the ratio of a new side length to the corresponding original side length.
2
For perimeter: multiply the original perimeter by \( k \).
3
For area: multiply the original area by \( k^2 \).
4
Check your units. Perimeter stays in linear units; area is in squared units.

Example 1: Rectangle with \( k = 3 \)

Original
\( 4 \) cm
\( 2 \) cm
\( \to \)
\( k = 3 \)
 Scaled
\( 12 \) cm
\( 6 \) cm
\( P: 12 \text{ cm} \to 36 \text{ cm } (\times 3) \)
\( A: 8 \text{ cm}^2 \to 72 \text{ cm}^2 (\times 9) \)

Example 2: Triangle with \( k = \frac{1}{2} \)

Original
\( 8 \) cm
\( 6 \) cm
\( 10 \) cm
\( \to \)
\( k = \frac{1}{2} \)
 Scaled
\( 4 \) cm
\( 3 \) cm
\( 5 \) cm
\( P: 24 \to 12 \)
\( (\times \frac{1}{2}) \)
\( A: 24 \to 6 \)
\( (\times \frac{1}{4}) \)

Example 3: Circle with \( k = 4 \)

\( r = 2 \)
 Original
\( \to \)
\( k = 4 \)
\( r = 8 \)
 Scaled
\( C: 4\pi \to 16\pi \)
\( (\times 4) \)
\( A: 4\pi \to 64\pi \)
\( (\times 16) \)

Working Backwards

Sometimes you're given the scaled measurements and need to find \( k \). Or you know how the area changed and must figure out the scale factor.

1
From side lengths: \( k = \frac{\text{new side}}{\text{original side}} \).
2
From perimeters: \( k = \frac{\text{new perimeter}}{\text{original perimeter}} \).
3
From areas: \( k = \sqrt{\frac{\text{new area}}{\text{original area}}} \). Take the square root because area scales by \( k^2 \).

Example 4: Finding \( k \) from Area

A square has an area of \( 25 \text{ cm}^2 \). A similar square has an area of \( 225 \text{ cm}^2 \). What is the scale factor?

1
Set up the ratio: \( \frac{\text{new area}}{\text{original area}} = \frac{225}{25} = 9 \).
2
Since area scales by \( k^2 \): \( k^2 = 9 \).
3
Solve: \( k = \sqrt{9} = 3 \).

Common Mistakes

✕ Using \( k \) for Area
Multiplying area by \( k \) instead of \( k^2 \). If \( k = 3 \), area increases by 9, not 3.
✕ Squaring Perimeter
Using \( k^2 \) for perimeter. Perimeter is one-dimensional: it only multiplies by \( k \).
✕ Forgetting \( \sqrt{} \)
Given an area ratio, forgetting to take the square root to find \( k \). If areas have ratio 16, then \( k = 4 \), not 16.

Summary Table

Scale Factor \( k \) Perimeter Multiplier Area Multiplier
\( 2 \) \( 2 \) \( 4 \)
\( 3 \) \( 3 \) \( 9 \)
\( 5 \) \( 5 \) \( 25 \)
\( \frac{1}{2} \) \( \frac{1}{2} \) \( \frac{1}{4} \)
\( \frac{1}{3} \) \( \frac{1}{3} \) \( \frac{1}{9} \)
\( 10 \) \( 10 \) \( 100 \)

Question 1 of 5

A rectangle has a perimeter of \( 20 \text{ cm} \). If the rectangle is scaled by a factor of \( k = 4 \), what is the new perimeter?

Question 2 of 5

A triangle has an area of \( 18 \text{ cm}^2 \). It is scaled by \( k = 3 \). What is the area of the new triangle?

Question 3 of 5

Two similar pentagons have areas of \( 16 \text{ m}^2 \) and \( 100 \text{ m}^2 \). What is the scale factor from the smaller to the larger?

Question 4 of 5

A square is scaled by \( k = \frac{1}{3} \). Its original area was \( 81 \text{ in}^2 \). What is the new area?

Question 5 of 5

A figure is scaled so its perimeter goes from \( 30 \text{ cm} \) to \( 75 \text{ cm} \). By what factor does the area change?