Enlargement and Scale Factor

What is an enlargement?

An enlargement is a transformation that changes the size of a shape. It is controlled by two things:

• A scale factor — how many times bigger (or smaller) the shape becomes.
• A centre of enlargement — a fixed point that the enlargement grows out of (or shrinks toward).

The image (the new shape) is always similar to the object (the original): same shape, all angles unchanged, every length multiplied by the scale factor.

The ray method

The reliable way to enlarge a shape is the ray method:

1. Draw a straight line (a ray) from the centre of enlargement through each corner of the object.
2. Measure how far each corner is from the centre.
3. Multiply that distance by the scale factor to find where the new corner goes along the same ray.
4. Join the new corners to draw the image.

Described diagram: picture a centre of enlargement marked with a dot at the origin. A small triangle sits nearby. From the centre, faint rays pass through each triangle corner and continue outward. With scale factor 2, each corner's new position is twice as far along its ray, producing a triangle twice the size, the same way up.

Worked example: whole-number scale factor

A vertex of a shape is 3 right and 1 up from the centre of enlargement. The scale factor is 3.

• New horizontal distance: 3 × 3 = 9 right
• New vertical distance: 1 × 3 = 3 up

So this corner's image is 9 right and 3 up from the centre. Repeat for every vertex, then connect them. The image is 3 times bigger and sits 3 times farther from the centre.

Fractional scale factors

A scale factor between 0 and 1 makes the shape smaller — it is still called an enlargement in maths, even though it shrinks.

With scale factor ½, a corner that is 6 right and 4 up from the centre moves to:

• 6 × ½ = 3 right
• 4 × ½ = 2 up

The image is half the size and closer to the centre.

Negative scale factors

A negative scale factor does two things at once:

• It scales the size by the number part (so −2 doubles the size).
• It sends the image to the opposite side of the centre, turning it upside down (rotated 180°).

A corner 2 right and 1 up from the centre, under scale factor −2, lands at:

• 2 × −2 = 4 left
• 1 × −2 = 2 down

Finding the centre and scale factor

If you are given an object and its image, you can work backwards:

• Scale factor = image length ÷ object length (matching sides).
• Centre = draw rays through matching corners of the object and image; where the rays cross is the centre of enlargement.

Worked example: a side of the object is 2 cm and the matching side of the image is 6 cm. Scale factor = 6 ÷ 2 = 3.

A note on area

Enlarging lengths by a scale factor of k multiplies the area by k². A scale factor of 3 makes the area 9 times bigger. Designers and map-makers rely on this when scaling plans up or down.

Activity: map your room

Measure your bedroom and draw a scale plan using a scale factor of, say, 1/50 (so 1 cm on paper represents 50 cm in real life). Mark the furniture. Then redraw it at twice the size to see enlargement in action — every length doubles, but the layout stays the same shape.

Where this connects

Enlargement is one of the four transformations, alongside the others in rotation and translation and symmetry and reflection. Because it produces similar shapes, it links directly to congruence and similarity.