How do you calculate dilation coordinates?

For center (a,b) and scale factor k: new x = a + k(x-a), new y = b + k(y-b). When center is origin: new x = kx, new y = ky.

What happens when the scale factor is negative?

A negative scale factor flips the image to the opposite side of the center of dilation, creating a mirror-like transformation.

Does dilation preserve angles?

Yes, dilation preserves all angle measures. The dilated figure is similar to the original.

How does dilation affect area?

Area is multiplied by k². A scale factor of 2 makes area 4 times larger; 0.5 makes it 1/4 the size.

What is the center of dilation?

The center of dilation is the fixed point from which all other points are scaled. It does not move during the transformation.

Dilation Calculator — Scale Factor & Coordinate Transformer

What Is Dilation in Geometry?

Dilation is a geometric transformation that changes the size of a figure without altering its shape. When you dilate a figure, every point moves toward or away from a fixed center point by a constant ratio called the scale factor. If the scale factor is greater than 1, the figure enlarges; if between 0 and 1, it shrinks.

The center of dilation serves as the anchor point. All distances from this center are multiplied by the scale factor to produce the new figure (the image). The original figure is called the pre-image.

The Dilation Formula

To perform a dilation with center point (a, b) and scale factor k, apply this formula to each point (x, y):

Dilation Formula:
(x', y') = (a + k(x - a), b + k(y - b))

When center is origin (0,0):
(x', y') = (kx, ky)

This formula works by finding the vector from the center of dilation to each point, multiplying that vector by the scale factor, and then translating back to get the final coordinates.

Types of Scale Factors

The scale factor determines how the figure changes:

k > 1: Enlargement — image is larger than pre-image
k = 1: Identity — image equals pre-image
0 < k < 1: Reduction — image is smaller
k < 0: Image flips to opposite side of center

Properties of Dilation

Dilations preserve several important geometric properties:

Angle measures remain unchanged
Parallel lines remain parallel
Shape is preserved (similar figures)
Ratios of distances are preserved

However, dilations change: lengths (multiplied by |k|), areas (multiplied by k²), and perimeters (multiplied by |k|).

Real-World Applications

Dilation appears in many contexts: map scaling, photography zoom, architectural scale models, computer graphics resizing, and biological growth patterns. Understanding dilation helps in fields ranging from art to engineering.

How to Use This Calculator

Enter the original coordinates (x, y), the center of dilation (a, b), and the scale factor (k). Results update in real time as you type, showing the new coordinates and transformation details.

Worked Examples

Example 1: Dilate (3, 4) from origin with k=2: (6, 8).

Example 2: Dilate (6, 9) from center (2, 1) with k=0.5: (4, 5).

Example 3: Dilate (4, -2) from center (1, 1) with k=3: (10, -8).

Dilation Calculator