Where do points go in a dilation?
Type the scale factor, the centre of dilation and the points of your figure. The dilation calculator moves each point to P′ = C + k(P − C), so (x, y) becomes (kx, ky) about the origin, shows the working, and says whether the figure grows or shrinks.
- Image points
- A′(−8, 12); B′(4, 2); C′(10, 8)
The image points are A′(−8, 12); B′(4, 2); C′(10, 8).
- Type of dilation
- Enlargement
- Length factor
- 2
- Area factor
- 4
- Working
- centre (0, 0), k = 2; A(−4, 6) → (0 + 2 × (−4 − 0), 0 + 2 × (6 − 0)) = A′(−8, 12); B(2, 1) → (0 + 2 × (2 − 0), 0 + 2 × (1 − 0)) = B′(4, 2); C(5, 4) → (0 + 2 × (5 − 0), 0 + 2 × (4 − 0)) = C′(10, 8)
Image points: A′(−8, 12); B′(4, 2); C′(10, 8). The image points are A′(−8, 12); B′(4, 2); C′(10, 8).
How it is worked out
How to calculate
Dilates points in the coordinate plane by a scale factor k about any centre, P′ = C + k(P − C), and says whether the figure is enlarged or reduced, with the length and area factors.
Example with the default inputs (Scale factor k 2, Centre x 0, Centre y 0, Points [x -4, y 6; x 2, y 1; x 5, y 4]): The image points are A′(−8, 12); B′(4, 2); C′(10, 8).
Method: P′ = C + k(P − C): x′ = cx + k(x − cx), y′ = cy + k(y − cy); lengths × |k|, areas × k².
- Typed decimals are read exactly, so each image coordinate is the exact decimal, rounded once to 10 significant figures.
- A negative scale factor puts the image on the other side of the centre, the same as a dilation by |k| and a half turn.
Worked examples
Each example is checked against the calculator on every build.
- Scale factor k 2, Centre x 0, Centre y 0, Points -4 6 gives Image points A′(−8, 12), Type of dilation Enlargement, Area factor 4.Source: CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)
- Scale factor k 0.5, Centre x 0, Centre y 0, Points 9 -13 gives Image points A′(4.5, −6.5), Type of dilation Reduction, Length factor 0.5, Area factor 0.25.Source: CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)
- Scale factor k 3, Centre x 1, Centre y 2, Points 2 3; 4 2; 1 5 gives Image points A′(4, 5); B′(10, 2); C′(1, 11), Area factor 9.Source: OpenStax, Elementary Algebra 2e, §8.7 Solve Proportion and Similar Figure Applications (similar figures have corresponding sides in the same ratio), https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications (retrieved 2026-10-02)
- Scale factor k -0.5, Centre x 0, Centre y 0, Points 4 -2 gives Image points A′(−2, 1), Type of dilation Reduction, turned 180° about the centre, Area factor 0.25.Source: CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)
How it works
A dilation with centre C = (cx, cy) and scale factor k sends each point P = (x, y) to
- x′ = cx + k(x − cx)
- y′ = cy + k(y − cy)
About the origin this is (x, y) → (kx, ky). Then:
- type: an enlargement when |k| > 1, a reduction when |k| < 1, the same size when |k| = 1; a negative k adds ", turned 180° about the centre"
- length factor = |k|, area factor = k²
The points are named A, B, C … in the order typed, and their images A′, B′, C′ ….
Each typed number is read as the exact decimal you typed, so every image coordinate is exact (sums and products of decimals are decimals). Each coordinate is then shown rounded half up to 10 significant figures.
Rules
- The scale factor is from −1,000,000 to 1,000,000 and not 0.
- Every coordinate, and the centre, is from −10¹² to 10¹². The centre is (0, 0) when left empty.
- 1 to 12 points.
Output format. Image points as A′(x, y); B′(x, y) with the true minus sign (−). The working lists centre (cx, cy), k = k, then for each point A(x, y) → (cx + k × (x − cx), cy + k × (y − cy)) = A′(x′, y′), each number to 10 significant figures. The length and area factors to 10 significant figures.
Worked examples by hand
A(−4, 6), k = 2, about the origin (CK-12). A′ = (2 × −4, 2 × 6) = (−8, 12), an enlargement; areas × 4.
A(9, −13), k = ½ (CK-12). A′ = (4.5, −6.5), a reduction; lengths × 0.5, areas × 0.25.
Triangle (2, 3), (4, 2), (1, 5), k = 3 about (1, 2). A′ = (1 + 3 × 1, 2 + 3 × 1) = (4, 5); B′ = (1 + 3 × 3, 2 + 3 × 0) = (10, 2); C′ = (1 + 3 × 0, 2 + 3 × 3) = (1, 11). Areas × 9.
A(4, −2), k = −0.5. A′ = (−0.5 × 4, −0.5 × −2) = (−2, 1): a reduction, turned 180° about the centre.
Other questions people ask
What is a dilation in geometry?
A dilation resizes a figure about a fixed point, the centre of dilation, by a scale factor k. Each image point lies on the line from the centre through the original point, k times as far from the centre. The image has the same shape: it is similar to the original.
How do I dilate a point about the origin?
Multiply both coordinates by the scale factor: (x, y) → (kx, ky). A(−4, 6) dilated by k = 2 is A′(−8, 12), and A(9, −13) dilated by k = ½ is A′(4.5, −6.5).
How do I dilate about a point that is not the origin?
Subtract the centre, multiply by k, and add the centre back: x′ = cx + k(x − cx) and y′ = cy + k(y − cy). With centre (1, 2) and k = 3, the point (2, 3) goes to (1 + 3 × 1, 2 + 3 × 1) = (4, 5).
Is it an enlargement or a reduction?
If |k| is more than 1 the image is bigger (an enlargement); if |k| is between 0 and 1 it is smaller (a reduction); if |k| = 1 it is the same size. A negative k also turns the image half a turn about the centre.
What happens to lengths and areas?
Every length is multiplied by |k| and every area by k². Dilating a triangle by 3 makes each side 3 times as long and the area 9 times as large. Angles do not change.
Can the scale factor be 0?
No. With k = 0 every point lands on the centre, so the image is a single point and not a figure. The calculator asks for a k other than 0.