How to Calculate the Scale Factor of a Dilation Guide

A dilation is a geometric transformation that enlarges or shrinks a figure while preserving its shape and the proportional relationships between all of its sides and angles. The scale factor (k) is the number that describes exactly how much the figure has changed in size: it is the ratio of any length in the new figure (the image) to the corresponding length in the original figure (the preimage). Finding it requires comparing one pair of corresponding measurements, either side lengths or distances from the centre of dilation to corresponding points.

What Is a Scale Factor in Geometry?

In geometry, the scale factor of a dilation is a constant ratio k that describes the relationship between every pair of corresponding lengths in the image and preimage. Because a dilation preserves shape (all angles remain equal, all sides change by the same factor), you only need one pair of corresponding lengths to find k. Every other pair of corresponding sides will produce the same ratio.

The three cases:

• k > 1: Enlargement. The image is larger than the preimage. A scale factor of 3 triples all linear dimensions.
• 0 < k < 1: Reduction. The image is smaller than the preimage. A scale factor of 0.5 halves all linear dimensions.
• k < 0: Negative dilation. The image appears on the opposite side of the centre of dilation and is also reflected through that point. This is less common in introductory geometry but appears in coordinate geometry problems.

Method 1: Scale Factor from Side Lengths

When the side lengths of both the image and preimage are known, the scale factor is found by dividing any image side length by the corresponding preimage side length.

Formula: k = Image side length ÷ Preimage side length

One condition: Always divide image by preimage, not preimage by image. The ratio from preimage to image gives the scale factor k. The ratio from image to preimage gives 1/k.

Side-Length Worked Example

Problem: Triangle ABC has sides of 3 cm, 4 cm, and 5 cm. After a dilation, triangle A’B’C’ has sides of 6 cm, 8 cm, and 10 cm. Find the scale factor.

Step 1: Identify one pair of corresponding sides Preimage side AB = 3 cm Image side A’B’ = 6 cm

Step 2: Divide image by preimage k = 6 ÷ 3 = 2

Verification, Check with a second pair: BC = 4 cm, B’C’ = 8 cm. k = 8 ÷ 4 = 2. Consistent.

Result: The scale factor is k = 2. The triangle was enlarged to twice its original size.

Use the dilation calculator to calculate coordinates and scale factors for geometric dilations.

Method 2: Scale Factor from Coordinates

When the coordinates of the centre of dilation and corresponding points are given, calculate the distance from the centre to each corresponding point, then divide the image distance by the preimage distance.

Simplified case, centre at the origin (0, 0): If the centre of dilation is the origin, divide any image coordinate by the corresponding preimage coordinate: k = x-image ÷ x-preimage (or y-image ÷ y-preimage)

General case, centre not at the origin: Use the distance formula for each point from the centre, then divide.

Coordinate Worked Example (Centre at Origin)

Problem: Point P is at (2, 3). Its image P’ after a dilation centred at the origin is at (5, 7.5). Find the scale factor.

Step 1: Divide the x-coordinates k = 5 ÷ 2 = 2.5

Step 2: Verify with y-coordinates k = 7.5 ÷ 3 = 2.5. Consistent.

Result: The scale factor is k = 2.5.

Coordinate Worked Example (Centre Not at Origin)

Problem: The centre of dilation is at C = (1, 1). Preimage point P = (3, 5). Image point P’ = (5, 9). Find the scale factor.

Step 1: Find the vector from centre to preimage (3 − 1, 5 − 1) = (2, 4)

Step 2: Find the vector from centre to image (5 − 1, 9 − 1) = (4, 8)

Step 3: Divide image vector component by preimage vector component k = 4 ÷ 2 = 2 (or k = 8 ÷ 4 = 2)

Result: The scale factor is k = 2.

How Does Scale Factor Affect Area and Volume?

Scale factor affects linear measurements, area, and volume differently. This distinction is a source of common errors.

Measurement	Effect of Scale Factor k
Length / Perimeter	Multiplied by k
Area	Multiplied by k²
Volume	Multiplied by k³

For the triangle enlarged by k = 2: the perimeter doubles (to 24 cm), but the area quadruples (2² = 4). If the original area was 6 cm², the new area is 24 cm². This is why real-world scale models require careful interpretation: a scale model car at 1:10 scale has sides that are 10 times smaller, but its cross-sectional area is 100 times smaller and its volume is 1,000 times smaller.

Sources

• Effortless Math: Scale Factor of a Dilation, definition, formula, enlargement vs. reduction, and worked examples.
• Khan Academy: Dilations and Scale Factors, coordinate-plane dilation exercises and conceptual explanation.

See all geometry and algebra tools in the maths calculators hub.