Dilation Calculator

Enter a point's X and Y coordinates, a scale factor (k), and the center of dilation (Cx, Cy) to find the transformed image point. The Dilation Calculator applies the formulas x' = Cx + k(x − Cx) and y' = Cy + k(y − Cy), returning the new coordinates (x', y') along with a breakdown of the transformation — whether it's an enlargement, reduction, or reflection.

The x-coordinate of the original point.

The y-coordinate of the original point.

A factor > 1 enlarges, 0 < k < 1 reduces, negative k reflects and scales.

The x-coordinate of the center of dilation.

The y-coordinate of the center of dilation.

Results

New X Coordinate (x')

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New Y Coordinate (y')

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Type of Dilation

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Distance from Center (Original)

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Distance from Center (Image)

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Original vs Image Coordinates

Frequently Asked Questions

What is dilation in geometry?

Dilation is a geometric transformation that changes the size of a figure without altering its shape. The resulting image is similar to the original — all angles are preserved and corresponding sides are proportional. The transformation is controlled by a scale factor and a fixed center of dilation.

What is the formula used to calculate dilation?

The dilation formulas are x' = Cx + k(x − Cx) and y' = Cy + k(y − Cy), where (x, y) is the original point, (Cx, Cy) is the center of dilation, and k is the scale factor. When the center is the origin (0, 0), this simplifies to x' = kx and y' = ky.

What is the scale factor in dilation?

The scale factor k determines how much the figure is enlarged or reduced. If k > 1, the image is an enlargement. If 0 < k < 1, the image is a reduction. If k = 1, the image is identical to the original. A negative scale factor also reflects the figure through the center of dilation.

How does a scale factor of ½ differ from a scale factor of 6?

A scale factor of ½ produces a reduction — the image is half the distance from the center of dilation compared to the original. A scale factor of 6 produces a significant enlargement, placing each image point six times farther from the center than the original point was.

What happens when the center of dilation is not at the origin?

When the center of dilation is not at the origin, you must account for its coordinates using the full formula x' = Cx + k(x − Cx). The point is effectively shifted relative to the center, scaled, and then shifted back. This is why entering the correct center coordinates matters greatly for accurate results.

Can the scale factor be negative?

Yes. A negative scale factor means the image appears on the opposite side of the center of dilation from the original point. For example, with k = −2, the image is twice as far from the center as the original, but in the opposite direction, producing both a reflection and an enlargement.

What is the difference between enlargement and reduction in dilation?

Enlargement occurs when the absolute value of the scale factor is greater than 1, making the image larger than the original. Reduction occurs when the absolute value is between 0 and 1, making the image smaller. In both cases, the shape remains geometrically similar to the original.

How do you find the scale factor given original and image coordinates?

To find the scale factor, calculate the distance from the center of dilation to the image point and divide it by the distance from the center to the original point: k = distance(center, image) ÷ distance(center, original). You can also use the ratio of corresponding coordinate differences: k = (x' − Cx) ÷ (x − Cx).

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