Geometry Tool
Dilation Calculator
Calculate dilated coordinates for any set of points given a center of dilation and scale factor. Shows formulas and classifies the transformation type.
Multiple Points-Custom Center-Step-by-Step-Enlargement / Reduction
Dilation Calculator
Points
A
B
C
Enlargementk = 2
| Point | Original (x, y) | Dilated (x', y') | Formula |
|---|---|---|---|
| A | (1, 2) | (2, 4) | (0 + 2(1 - 0), 0 + 2(2 - 0)) |
| B | (3, 4) | (6, 8) | (0 + 2(3 - 0), 0 + 2(4 - 0)) |
| C | (5, 1) | (10, 2) | (0 + 2(5 - 0), 0 + 2(1 - 0)) |
Dilation Formula
P'(x', y') = (Cx + k(Px - Cx), Cy + k(Py - Cy))
Center: (0, 0), Scale factor: 2
Scale Factor Effects
| Scale Factor (k) | Type | Size Change | Area Change |
|---|---|---|---|
| k > 1 | Enlargement | Larger | k^2 times larger |
| k = 1 | Congruent | Same size | No change |
| 0 < k < 1 | Reduction | Smaller | k^2 times smaller |
| k = 0 | Collapse | Point at center | Zero |
| -1 < k < 0 | Reflected + Reduction | Smaller, reflected | k^2 times smaller |
| k < -1 | Reflected + Enlargement | Larger, reflected | k^2 times larger |
Frequently Asked Questions
What is a dilation in geometry?⌄
A dilation is a transformation that changes the size of a figure without changing its shape. It is defined by a center point and a scale factor k. Every point moves closer to or farther from the center by factor k. The resulting figure is similar to the original.
What is the formula for dilation?⌄
For center (Cx, Cy) and scale factor k: P'(x', y') = (Cx + k(Px - Cx), Cy + k(Py - Cy)). If center is the origin: P'(x', y') = (k*Px, k*Py).
What does a scale factor greater than 1 mean?⌄
Scale factor k > 1 means enlargement - the image is larger than the original. k = 2 doubles all distances from the center. k = 3 triples them.
What does a scale factor between 0 and 1 mean?⌄
Scale factor 0 < k < 1 means reduction - the image is smaller. k = 0.5 halves all distances from the center, creating a figure half the original size.
What does a negative scale factor do?⌄
A negative scale factor reflects the figure through the center point AND scales it. k = -2 reflects through the center and doubles size. The image appears on the opposite side of the center.
Does dilation preserve angles?⌄
Yes. Dilation preserves all angles, parallelism, and the shape of the figure. It only changes size (lengths and areas). The original and image are similar figures.
How does dilation affect area?⌄
Area scales by k squared. If k = 3, the area of the image is 9 times the original. If k = 0.5, the area is 0.25 (one quarter) of the original.
What is the center of dilation?⌄
The fixed point from which all points are scaled. It does not move during the transformation. If the center is at the origin (0,0), the formula simplifies to just multiplying each coordinate by k.