How to Find the Scale Factor of a Dilation: Sides & Coordinates

To find a positive dilation scale factor, divide an image length by its corresponding original length: k = image length ÷ original length. For example, a side changing from 6 units to 9 units gives k = 9 ÷ 6 = 1.5. Match the sides first so you do not reverse the ratio.

How to find the scale factor of a dilation

A dilation resizes a figure about a fixed point called its center. The original is the preimage; the transformed figure is the image. Corresponding angles remain equal, and corresponding lengths share one scale ratio.

1. Identify the original and image. Vertex A corresponds to A′, so side AB corresponds to A′B′.
2. Choose matching lengths in the same unit. Use the given measurements, not how large a sketch looks.
3. Divide image by original. If AB = 8 and A′B′ = 12, then k = 12 ÷ 8 = 1.5 for a positive dilation.
4. Check another corresponding pair. If BC = 10, its image must be 15 for the same factor. Unequal ratios cannot describe one dilation.

For positive k, k > 1 enlarges the figure, 0 < k < 1 reduces it, and k = 1 leaves it unchanged. Going in the reverse direction uses the reciprocal: changing from 12 units back to 8 gives 8 ÷ 12 = 2/3.

Find a new length: multiply the original by k. A square with side 6 units dilated by k = 3 has image side 6 × 3 = 18 units. To recover an original length, divide the image length by k. These length formulas assume a positive factor; the signed case is explained below.

Use the center of dilation correctly

With positive k, each point and its image lie on the same ray from the center. The image’s distance from the center is k times the original distance. In this diagram, R stays fixed and RA′ = 2RA, RB′ = 2RB, and RC′ = 2RC.

You can find k from corresponding distances to the center, provided the original point is not the center itself. If RP = 4 and RP′ = 10 on the same ray, k = 10 ÷ 4 = 2.5. The center cannot supply a ratio because its distance is zero.

How to find the scale factor from coordinates

When the center is the origin

The coordinate rule is (x, y) → (kx, ky). For P(4, 6) → P′(6, 9), divide matching nonzero coordinates: 6 ÷ 4 = 1.5 and 9 ÷ 6 = 1.5. Both checks agree.

If a coordinate is zero, use the other coordinate instead. P(0, 4) → P′(0, 10) gives k = 10 ÷ 4 = 2.5; 0 ÷ 0 is undefined. One point pair gives a candidate factor; check all given pairs against the same center and rule.

When the center is not the origin

Subtract the center coordinates before comparing. For center C(a, b), the rule is x′ = a + k(x − a) and y′ = b + k(y − b).

Example: C(1, 2), P(3, 5), and P′(5, 8). The original displacement from C is (2, 3); the image displacement is (4, 6). Thus k = 4 ÷ 2 = 6 ÷ 3 = 2. Dividing the raw coordinates 5 ÷ 3 or 8 ÷ 5 would be wrong.

What if the scale factor is negative?

Some courses also use signed factors. A negative k puts a point’s image on the opposite side of the center. Lengths and distances multiply by |k|, the absolute value of k; directed coordinate displacements multiply by k itself. Ordinary side lengths alone reveal the magnitude, not the sign.

About the origin, P(2, 3) → P′(−4, −6) gives k = −2. The image is twice as large and lies across the center, equivalent to a factor-2 enlargement followed by a 180° rotation about the center. For k = −1, the size is unchanged. A factor of zero collapses every point to the center and is excluded from the nondegenerate dilations discussed here.

Area is different from length: area multiplies by k². A factor of 2 doubles side lengths but quadruples area. A factor of 1/2 halves lengths but gives 1/4 of the original area.

Practice: find the factor or missing length

1. A square has original side 6 units. It is dilated by positive k = 3. Find the image side.
2. A square’s side changes from 3 units to 6 units in a positive dilation. Find k.
3. About the origin, P(−2, 4) becomes P′(−1, 2). Find k.
4. With center C(1, 2), P(3, 5) becomes P′(5, 8). Find k.
5. About the origin, P(2, 3) becomes P′(−4, −6). Find the signed factor and the length multiplier.

Further learning and practice

Next, practice graphing dilations on the coordinate plane. For a broader sequence of free lessons, use the Geometry course.

For a book option, open Geometry for Beginners and check its preview before choosing it for your review.

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Related lessons: performing dilations · similar figures · similarity and ratios.

References: Common Core dilation length-ratio standard; Netmath mathematical lexicon: dilation.

What the scale factor tells you

Once you have k, you can tell what kind of dilation it is without looking at a picture.

Lengths are multiplied by |k| and areas by k². So with k = 3, a shape’s area becomes 9 times as large.

Example: dilate by a scale factor of 1/3

Triangle ABC has A(3, 6), B(9, 0) and C(0, -3). Dilate it by k = 1/3 with the center at the origin.

1. Multiply each coordinate by 1/3.
2. A(3, 6) becomes A'(1, 2).
3. B(9, 0) becomes B'(3, 0).
4. C(0, -3) becomes C'(0, -1).

Check with one side. AB runs from (3, 6) to (9, 0), so its length is √(6² + 6²) = √72 ≈ 8.49. A’B’ runs from (1, 2) to (3, 0), so its length is √(2² + 2²) = √8 ≈ 2.83. Then 2.83 ÷ 8.49 ≈ 0.33, which is 1/3. The image is a reduction, one third the size of the original.

Frequently Asked Questions

What is the formula for the scale factor of a dilation?

The formula is k = image length ÷ pre-image length. Use any pair of matching sides or matching distances from the center. For example, if a side of 6 becomes 15, k = 15 ÷ 6 = 2.5. With the center at the origin, you can also divide a new coordinate by the matching old one, as long as the old coordinate is not 0.

How do you dilate a figure by a scale factor of 1/3?

If the center is the origin, multiply both coordinates of every vertex by 1/3. For example, (6, -9) becomes (2, -3). Then connect the new points. Every side of the image will be one third as long as before. If the center is another point, multiply each point’s distance from the center by 1/3 instead.

How do you find the scale factor from coordinates?

With the center at the origin, divide an image coordinate by the matching original coordinate. If P(4, 6) maps to P'(10, 15), then k = 10 ÷ 4 = 2.5, and 15 ÷ 6 = 2.5 confirms it. If the center is not the origin, measure each point’s distance from the center and divide those distances.

How can you tell if a dilation is an enlargement or a reduction?

Look at the size of k. If k is greater than 1, the image is larger, so it is an enlargement. If k is between 0 and 1, the image is smaller, so it is a reduction. If k equals 1, the image is the same size as the original.

Can a scale factor be negative?

Yes. A negative scale factor puts the image on the opposite side of the center and turns it 180°. Lengths are still multiplied by the absolute value of k. For example, with k = -2 and center at the origin, (2, 3) maps to (-4, -6), and every side becomes twice as long.