Enlargement

Scaling a shape from a centre

An enlargement resizes a shape. It is defined by two things: a centre of enlargement — a fixed point everything is measured from — and a scale factor k, the number every length is multiplied by.

To find where a point goes, measure its distance from the centre and multiply that distance by k. Every length in the shape is multiplied by k too, but the angles are unchanged — so the image is a scaled copy: it is similar to the original.

k > 1 makes the shape bigger;
0 < k < 1 makes it smaller.

What changes, and by how much

Lengths scale by k, but area scales differently — a shape is two dimensions wide, and each one stretches by k.

Under an enlargement with scale factor k:

every length is multiplied by k;
angles are unchanged, so the image is similar to the original;
area is multiplied by k^2 — two dimensions each scale by k;
a fractional scale factor (0 < k < 1) shrinks the shape.

See the enlargement

Step through an enlargement of a small triangle from the centre (0, 0) with scale factor k = 2. Each vertex moves twice as far from the centre along a ray, so the image is twice the size with the same angles.