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Dilation Calculator

Scale a point from the origin or from a custom center, then see the new x and y values for any scale factor in geometry work.

Result

(6, 8)

Dilating (3, 4) by a scale factor of 2 about (0, 0) gives (6, 8)

Quick Answer

A dilation calculator applies a coordinate transformation from a fixed center. About the origin, (3, 4) with k = 2 becomes (2 × 3, 2 × 4) = (6, 8). For a custom center, subtract its coordinates, scale the horizontal and vertical offsets, then add the center back. Negative k reverses direction through the center.

Dilation Calculator Explained: Center, Scale And Points

A dilation moves a point on a ray from a fixed center. The scale factor tells how far the new point sits from that center. A value above one makes the gap grow. A value from zero to one makes the gap shrink. When the center is (0, 0), multiply both point values by k. A custom center needs one more step. First find the x and y gaps from the center. Scale those gaps. Then add the center back. A negative k puts the image on the other side of the center. The size of k still sets the new gap. A zero factor sends the point to the center. This tool works one point at a time, so each step stays clear.

The Dilation Formula And How Each Coordinate Is Found

Dilation maps coordinates by applying a scale factor to horizontal and vertical offsets relative to the center of dilation.

  • x' = cx + k(x − cx).
  • y' = cy + k(y − cy).
  • Origin centering (cx = 0, cy = 0): (x', y') = (kx, ky).

x and y: coordinates of the original point.

cx and cy: coordinates of the center of dilation.

k: scale factor applied to both offsets.

x' and y': coordinates of the image point.

Using This Coordinate Dilation Tool In Clear Steps

Inputs

  • Point x: enter the horizontal coordinate of the original point.
  • Point y: enter the vertical coordinate of the original point.
  • Scale factor: enter k to multiply the distance from the center.
  • Center x: leave at zero for origin-centered dilation.
  • Center y: leave at zero for origin-centered dilation.

Steps

  1. Choose origin mode or use the custom-center fields.
  2. Enter the original point.
  3. Enter the scale factor k.
  4. Enter the center coordinates when the center is not the origin.
  5. Calculate and compare the old and new offsets from the center.

Coordinate Dilation Example With Real Number Steps Shown

Original point (3, 4), center of dilation (0, 0), scale factor k = 2.

  1. Find the horizontal offset: x − cx = 3 − 0 = 3.
  2. Scale and restore x: x' = cx + k(x − cx) = 0 + 2 × 3 = 6.
  3. Find the vertical offset: y − cy = 4 − 0 = 4.
  4. Scale and restore y: y' = cy + k(y − cy) = 0 + 2 × 4 = 8.

The dilated point is (6, 8). Exact whole-number arithmetic is used, so no rounding is needed. For custom center (1, 1) and point (4, 3) with k = 2, the image point is (7, 5).

Where A Coordinate Dilation Tool Helps In Geometry Class

Use this tool to check a point in a geometry change, a similar-shape task, or a grid problem. Use custom-center mode when the marked center is not the origin.

For other coordinate geometry, transformations, and grid calculation tools, visit the math calculators hub and look up concepts in the glossary.

Assumptions

  • One scale factor applies to both coordinate directions.
  • The center stays fixed during the transformation.
  • Coordinates use the same unit and axis scale.
  • A negative factor includes placement on the opposite side of the center.

Limitations

  • The tool transforms one point at a time.
  • A full shape requires entering each vertex separately.
  • The calculator does not draw construction rays or a full coordinate grid.
  • Some school texts define a geometric dilation with positive k only.

In Practice

Do not scale the raw point values when the center is not zero. Find x - cx and y - cy first. Multiply those gaps by k, then add the center back.

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Related Guides

Dilation Calculator Questions And Coordinate Answers

What Does A Scale Factor Do In A Dilation?

A dilation scale factor multiplies every point's distance from the fixed center. A value above one grows the figure. A value from zero to one shrinks it. Zero sends all points to the center, and a negative value moves points to the other side.

How Do You Dilate A Point About The Origin?

Dilate a point about the origin by multiplying both coordinates by one scale factor. A point (x, y) becomes (kx, ky). For (3, 4) with k = 2, the work is (2 × 3, 2 × 4) = (6, 8).

How Is A Custom Center Different From The Origin?

A custom center changes the coordinate offsets that the scale factor must multiply. Subtract the center from the point, multiply each gap by k, and add the center back. Direct point scaling works only when the center is the origin.

Is A Dilation The Same As A Translation?

A dilation and a translation are different coordinate transformations with different effects. A shift, or translation, adds the same move to every point. A dilation can change size, while a shift keeps the same size and shape.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.