The Shape Dilation Calculator calculates the coordinates of a shape after it is enlarged or reduced by a scale factor k from a center point.
Each point moves along the line from the center using x′ = cx + k(x − cx) and y′ = cy + k(y − cy). Enter the points, the scale factor and the center of dilation (the origin by default); the calculator lists the image points, draws both shapes and shows the area scale factor k².
Shape Dilation Calculator (Enlargement)
Dilate (enlarge or reduce) a shape by a scale factor about the origin or any centre, with the new coordinates, graph and area change.
Formula
x′ = cx + k(x − cx)
y′ = cy + k(y − cy)
About the origin this simplifies to (x, y) → (kx, ky).
Where:
- (x, y) = original point; (x′, y′) = image point
- (cx, cy) = center of dilation
- k = scale factor
Lengths scale by |k| and areas by k².
How the Calculation Works
Step 1: Find each point’s offset from the center: (x − cx, y − cy).
Step 2: Multiply the offset by k.
Step 3: Add the center back to get the image point.
Variables Explained
| Scale factor k | Effect |
|---|---|
| k > 1 | Enlargement |
| k = 1 | No change |
| 0 < k < 1 | Reduction |
| k < 0 | Resized and turned 180° through the center |
Example Calculation
Scale factor 2 about the origin
Given: A(1, 1), B(5, 1), C(2, 4); k = 2
Result: A′(2, 2), B′(10, 2), C′(4, 8); the area becomes 2² = 4 times larger
Scale factor 3 about (1, 1)
B(5, 1): offset (4, 0) × 3 = (12, 0), plus (1, 1) → B′(13, 1)
C(2, 4): offset (1, 3) × 3 = (3, 9), plus (1, 1) → C′(4, 10)
A(1, 1) is the center, so it stays at (1, 1).
Shape Properties
A dilation produces a similar shape: angles stay the same and every length is multiplied by the same factor. Unlike rotations, reflections and translations, a dilation changes size, so the image is not congruent unless k = ±1.
Dimensions Explained
Finding the scale factor
Divide an image length by the matching original length, or divide a point’s distance from the center after dilation by its distance before.
Finding the center
Draw lines through each point and its image; they meet at the center of dilation.
Frequently Asked Questions
What is the rule for a dilation about the origin?
(x, y) → (kx, ky).
How do you dilate a shape about a point other than the origin?
Use x′ = cx + k(x − cx) and y′ = cy + k(y − cy).
What happens with a scale factor between 0 and 1?
The shape shrinks toward the center.
What does a negative scale factor do?
It resizes the shape and places it on the opposite side of the center, as if rotated 180°.
How does dilation affect area?
Area is multiplied by k². Doubling the lengths makes the area four times larger.