What Is A Dilation In Math? Scale Factor And Examples
The short version
- A dilation is a transformation that makes a figure bigger or smaller by a scale factor, measured out from a fixed center point. The shape stays the same; only the size changes.
- Centered at the origin, the rule is (x, y) to (kx, ky). Multiply both coordinates by the scale factor k.
- If k is greater than 1, the figure grows. If k is between 0 and 1, it shrinks. If k equals 1, nothing changes.
- Lengths multiply by k, but area multiplies by k² and volume by k³. A dilation by 3 makes the area 9 times bigger.
- To find the scale factor, divide an image length by the matching original length.
A dilation in math is a transformation that resizes a figure by a scale factor while keeping its shape. Every point moves along a straight line away from (or toward) a fixed point called the center of dilation, and every distance from that center gets multiplied by the same number.
That’s the full dilation definition in geometry. The image has the same angles as the original and its sides stay in the same proportion, so the two figures are similar. Of the four basic transformations, dilation is the only one that changes size.
The words you need for dilation problems
| Term | What it means |
|---|---|
| Pre-image | The original figure, before the dilation |
| Image | The new figure, usually labeled with primes (A’, B’, C’) |
| Center of dilation | The fixed point everything stretches away from or shrinks toward. It doesn’t move. |
| Scale factor (k) | The number every distance from the center is multiplied by |
| Enlargement | A dilation with k greater than 1 |
| Reduction | A dilation with k between 0 and 1 |
One quick way to picture it: a projector. The bulb is the center of dilation. Move the projector farther from the wall and the picture gets bigger, but a circle on the slide is still a circle on the wall.
How to do a dilation centered at the origin
This is the version most homework starts with. Multiply each coordinate of each vertex by k, plot the new points, and connect them in order.
Example 1: enlarging a rectangle
Rectangle ABCD has A(1, 2), B(3, 2), C(3, 5), D(1, 5). Dilate by k = 2.
A(1, 2) to A'(2, 4)
B(3, 2) to B'(6, 4)
C(3, 5) to C'(6, 10)
D(1, 5) to D'(2, 10)
Check: the original is 2 wide and 3 tall. The image is 4 wide and 6 tall. Both doubled, so it’s the same shape at twice the size.
Example 2: shrinking a triangle
Triangle with vertices (4, 8), (8, 0), and (-4, 4). Dilate by k = 1/4.
(4, 8) to (1, 2)
(8, 0) to (2, 0)
(-4, 4) to (-1, 1)
Because 1/4 is less than 1, every point pulled in toward the origin.
Dilation with a center that isn’t the origin
This is where most kids get tripped up, and many guides skip it. You can’t just multiply the coordinates, because that measures from (0, 0) instead of from the real center. Instead, measure from the center, scale that distance, and add it back.
- Subtract the center from the point to get how far it sits from the center, left/right and up/down.
- Multiply both of those distances by k.
- Add the result back to the center.
As a formula, with center (a, b): (x, y) goes to (a + k(x – a), b + k(y – b)).
Example 3: center at (1, 1)
Dilate the point (4, 3) by k = 2 with center (1, 1).
Distance from center: 4 – 1 = 3 right, 3 – 1 = 2 up.
Scale it: 3 × 2 = 6 right, 2 × 2 = 4 up.
Add to the center: (1 + 6, 1 + 4) = (7, 5).
If you had multiplied the coordinates directly, you’d get (8, 6), which is wrong for this center.
How to find the scale factor
Divide a length in the image by the matching length in the original:
k = image length ÷ original length
A side that’s 6 cm in the original and 15 cm in the image gives k = 15 ÷ 6 = 2.5. It’s an enlargement.
With coordinates and the center at the origin, divide an image coordinate by the original. If (2, -3) maps to (8, -12), then k = 8 ÷ 2 = 4, and -12 ÷ -3 = 4 agrees. Always check both coordinates. If they give different numbers, it isn’t a dilation from the origin.
What a dilation does to perimeter, area and volume
This part shows up on almost every unit test, and it’s where a lot of points get lost.
| Measurement | Multiplied by | If k = 3 |
|---|---|---|
| Side lengths, perimeter | k | 3 times as long |
| Area | k² | 9 times as big |
| Volume | k³ | 27 times as big |
| Angles | unchanged | same angles |
Example 4: checking it with a rectangle
A 3 by 5 rectangle has perimeter 3 + 5 + 3 + 5 = 16 and area 3 × 5 = 15.
Dilate by 3 and it becomes 9 by 15.
New perimeter: 9 + 15 + 9 + 15 = 48, which is 16 × 3.
New area: 9 × 15 = 135, which is 15 × 9.
Area grows by k² because it’s length times width, and both of those got multiplied by k.
What about a negative scale factor?
Some courses include them. A negative k scales by the size of k and also sends each point to the opposite side of the center. With k = -2 from the origin, (3, 1) goes to (-6, -2). It’s the same as dilating by 2 and then rotating 180° around the center.
Common mistakes
- Adding instead of multiplying. Dilating (3, 4) by 2 gives (6, 8), not (5, 6).
- Multiplying coordinates when the center isn’t the origin. Measure from the actual center first.
- Using k for area. Doubling the sides of a square makes its area 4 times bigger, not 2.
- Flipping the fraction. The scale factor is image over original. Original over image gives the reverse dilation.
- Thinking any fraction means bigger. A scale factor of 3/2 enlarges; a scale factor of 2/3 shrinks. Compare it to 1.
- Calling the image congruent. It’s similar. Congruent figures are the same size, and a dilation (other than k = 1) changes size.
Practice problems
- Dilate (3, -2) by a scale factor of 4, centered at the origin.
- Dilate (10, 6) by a scale factor of 1/2, centered at the origin.
- A segment 8 cm long becomes 20 cm after a dilation. What’s the scale factor?
- Centered at the origin, (6, 9) maps to (4, 6). What’s the scale factor? Is it an enlargement or a reduction?
- A triangle with area 12 square units is dilated by a scale factor of 3. What’s the new area?
- Dilate (5, 2) by a scale factor of 3 with center (2, 1).
- Dilate (-2, 4) by a scale factor of -1/2, centered at the origin.
- A square with perimeter 20 is dilated, and the image has area 100. What was the scale factor?
Answers
- (12, -8)
- (5, 3)
- 20 ÷ 8 = 2.5
- 4 ÷ 6 = 2/3, and 6 ÷ 9 = 2/3 agrees. It’s a reduction because 2/3 is less than 1.
- 12 × 3² = 12 × 9 = 108 square units
- (11, 4). Distance from center is 3 right and 1 up; tripled, that’s 9 right and 3 up; (2 + 9, 1 + 3) = (11, 4).
- (1, -2). Multiply each coordinate by -1/2.
- 2. The original side is 20 ÷ 4 = 5 and the image side is 10, since 10 × 10 = 100. Then 10 ÷ 5 = 2.
If your child is stuck on dilations
Dilations are a geometry lesson built on older skills. The multiplication is easy when k is 2. It stops being easy when k is 2/3 or 2.5, and that’s where kids who never got solid with fractions and decimals fall apart. Finding a scale factor is really a ratio problem, and a ratio problem is a division problem.
If your child can do Example 1 but not problem 4, the issue probably isn’t geometry at all. Our breakdown of math skills by grade level shows when fraction skills are supposed to land, and why kids struggle with algebra explains how a fourth grade fraction gap keeps showing up years later. Short, regular work on the missing piece does more than a long cram session; here’s why 20 minutes a day works better.
Frequently asked questions
What is a dilation in geometry?
It’s a transformation that enlarges or shrinks a figure by a scale factor from a fixed center point. The image is similar to the original: same shape and angles, different size.
Is a dilation a rigid transformation?
No. Rigid transformations (translations, rotations, reflections) keep lengths the same. A dilation changes lengths, so it’s a similarity transformation instead.
How do you find the scale factor of a dilation?
Divide any length in the image by the matching length in the original. If the result is more than 1, it’s an enlargement; if it’s between 0 and 1, it’s a reduction.
Does a dilation change angles?
No. Every angle in the image matches the original. That’s why the shape looks the same, just resized. Parallel sides also stay parallel.
What happens to area when you dilate a figure?
Area is multiplied by the square of the scale factor. A scale factor of 2 makes the area 4 times as big, and a scale factor of 1/2 makes it 1/4 as big.
Find the actual gap
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