Enlargement

An enlargement is a type of geometric transformation that changes the size of a figure while preserving its shape. Unlike translations, rotations, and reflections — which are isometric transformations that keep lengths unchanged — an enlargement scales every length in the figure by a constant called the scale factor. The result is a figure that is mathematically similar to the original: all corresponding angles are equal, and all corresponding side lengths are in the same ratio. Every enlargement is defined by two things: the center of enlargement (a fixed point from which the scaling radiates) and the scale factor k. If k > 1, the image is larger than the original object. If 0 < k < 1, the image is smaller (sometimes called a reduction). If k < 0, the image appears on the opposite side of the center of enlargement and is also rotated by 180°. When k = 1 the transformation is the identity — the figure does not change at all. Enlargements appear throughout mathematics and the real world. Architects use scale drawings, photographers zoom lenses, biologists magnify specimens under microscopes, and engineers produce scaled blueprints — all of these rely on the mathematics of enlargement. Understanding how to perform and reverse enlargements is a core skill in Euclidean geometry and underpins the study of similar figures, trigonometric ratios, and coordinate geometry.

Key takeaways
  • Identify the center of enlargement and the scale factor
  • Draw rays from the center through each vertex
  • Measure and scale each distance
  • Connect the image vertices
  • Verify similarity and area scaling

Identify the center of enlargement and the scale factor

The center of enlargement O is a fixed point — it can lie inside, outside, or on the boundary of the original figure. The scale factor k tells you how much each distance from O is multiplied. For example, k = 3 means every point moves to three times its original distance from O.

Draw rays from the center through each vertex

For each vertex P of the original figure, draw a ray starting at O and passing through P. These rays are called lines of enlargement. Every image point lies somewhere along one of these rays, on the same side as P when k > 0, and on the opposite side when k < 0.

Measure and scale each distance

Measure the distance OP from the center O to each vertex P. Multiply this distance by the scale factor k to obtain the distance OP'. Mark P' on the ray at distance k × OP from O. Repeat for every vertex of the figure. When using coordinates, apply the formula directly: if O = (a, b) and P = (x, y), then P' = (a + k(x − a), b + k(y − b)).

Connect the image vertices

Join the image points P' in the same order as the original vertices. The resulting figure is the image under the enlargement. Check your work: every pair of corresponding sides should be parallel, and the ratio of image length to object length should equal |k| for every side.

Verify similarity and area scaling

The image is always similar to the object. All corresponding angles are equal. Corresponding lengths are in the ratio k : 1. Areas scale by the square of the scale factor: Area(image) = k² × Area(object). Volumes (in 3D) scale by k³. These relationships are critical for solving problems about similar figures.

The Key Equation

P′=(a+k(x−a),  b+k(y−b))P' = \bigl(a + k(x - a),\; b + k(y - b)\bigr)
  • P' = (x', y') — Image point (none): The coordinates of the transformed (enlarged) point
  • O = (a, b) — Center of enlargement (none): The fixed point from which all distances are scaled
  • k — Scale factor (none (dimensionless ratio)): Multiplies every distance from the center; k > 1 enlarges, 0 < k < 1 reduces, k < 0 enlarges and inverts
  • (x, y) — Object point P (none): The original coordinates of a vertex or point before transformation

Worked Example

Problem. Triangle ABC has vertices A(1, 1), B(4, 1), and C(1, 5). It is enlarged with center of enlargement O(0, 0) and scale factor k = 2. Find the coordinates of the image triangle A'B'C' and calculate the ratio of the area of A'B'C' to the area of ABC.

A′(2,2),  B′(8,2),  C′(2,10);Area(A′B′C′)Area(ABC)=k2=4A'(2,2),\; B'(8,2),\; C'(2,10);\quad \frac{\text{Area}(A'B'C')}{\text{Area}(ABC)} = k^2 = 4

Answer. A'(2, 2), B'(8, 2), C'(2, 10). The area of the image triangle is 4 times the area of the original triangle.

Figure
Diagram
xyABCA'B'C'O3648Scale factor k = 2Blue: original △ABCRed: image △A′B′C′
Origins · 1 minEuclid of Alexandria and how this was discovered
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Euclid of Alexandria
-325–-265 · Greek

Euclid's Elements (c. 300 BCE) established the axiomatic foundation of geometry. Book VI of the Elements develops the theory of similar figures and proportionality — the mathematical backbone of enlargements. His definition that similar polygons have proportional sides and equal corresponding angles is the geometric statement of what scale factor k captures algebraically. When King Ptolemy I asked Euclid if there was a shorter path to mastering geometry than studying the Elements, Euclid reportedly replied: 'There is no royal road to geometry.' Despite writing one of the most influential textbooks in history, almost nothing is known about Euclid's personal life.

  • Similar Triangles — Enlargement always produces a similar figure. Similar triangles have equal angles and sides in constant ratio — exactly what a scale factor k describes.
  • Scale Factor and Ratio — The scale factor k is the central numerical tool of enlargement; understanding ratios is essential for calculating image lengths, areas, and volumes.
  • Reflections — Reflections are another geometric transformation. Comparing isometric transformations (reflection, rotation, translation) with non-isometric ones (enlargement) deepens understanding of the full family of transformations.
  • Coordinate Geometry — Enlargements are often performed algebraically using coordinates. The formula P' = (a + k(x−a), b + k(y−b)) links transformation geometry directly to coordinate methods.