Enlargement
Making a shape bigger — or smaller
Set a photocopier to "enlarge 200%" and the page comes out twice as big in every direction —
same picture, just scaled up. Blowing a thumbnail up to a poster, or shrinking a whole room
onto a floor plan, is the same move: resizing a shape without distorting it.
An enlargement changes the size of a shape without changing its
shape. You pick a number called the scale factor
k, and every length is multiplied by it. A scale
factor of 2 doubles every side; a scale factor of
3 trebles every side; a scale factor of
\tfrac{1}{2} halves every side.
Because every length grows by the same amount, the new shape — called the
image — is a perfect scaled copy of the original. The corners sit at exactly
the same angles, so it looks identical, just a different size. Two shapes that are the same
shape but different sizes are called
similar.
- k > 1 makes the shape bigger (an enlargement);
- 0 < k < 1 makes it smaller (a reduction);
- k = 1 leaves it exactly the same.
When you pinch to zoom a photo on a phone, you are doing an enlargement. Every part of the
picture grows by the same scale factor at once, so the cat still looks like a cat — just
bigger. If the photo stretched only sideways, the cat would look squashed and wrong. An
enlargement keeps the proportions, which is exactly why the zoomed picture still looks
right.
→ zoom by 2 →
A scale factor of 2 doubles every side
Suppose a rectangle is 3 cm wide and 2 cm
tall, and we enlarge it by scale factor k = 2. We multiply
both the width and the height by 2:
3 \times 2 = 6 \text{ cm wide}, \qquad 2 \times 2 = 4 \text{ cm tall}.
The image is a 6 \times 4 rectangle — twice as wide and twice as
tall. It is still a rectangle (all its angles are still right angles); it is just bigger.
Worked example. A triangle has sides 4 cm,
5 cm and 6 cm. Enlarged by scale factor
3, the new sides are 4 \times 3 = 12 cm,
5 \times 3 = 15 cm and 6 \times 3 = 18
cm. Every side tripled, so the triangle keeps its shape.
The centre of enlargement
To put the image in a definite place, an enlargement needs a fixed point called the
centre of enlargement. You measure each corner's distance from the centre and
multiply that distance by the scale factor. So a corner that is
3 squares from the centre ends up 3 \times 2 = 6
squares from the centre, in the same direction.
A neat way to see it: draw a straight ray from the centre through each corner.
Every corner of the image slides out along its own ray to k times
the distance. Step through it below, from the centre (0, 0) with
k = 2.
See it: enlarge from the origin
Here is a rectangle (filled) and its image after enlarging from the centre at
(0, 0). The dashed rays show how each corner slides straight out to
k times its distance. Press Refresh for a new shape
and a new scale factor, and watch how the image grows.
What happens to the area?
Lengths scale by k, but area scales by
k^2. A shape is two dimensions wide, and each direction
stretches by k, so the area stretches by
k \times k = k^2.
Worked example. A 3 \times 2 rectangle has area
6 cm². Enlarged by k = 2 it becomes a
6 \times 4 rectangle with area
24 cm². That is 6 \times 2^2 = 6 \times 4 = 24
— the area went up four times, not twice. Treble the sides
(k = 3) and the area goes up 3^2 = 9 times.
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;
- a fractional scale factor (0 < k < 1) shrinks
the shape.
The two enlargement traps:
- Multiply every side by the scale factor — not just one. If you stretch
only the width, you get a squashed shape, not an enlargement.
- The angles do not change — only lengths do. A right angle stays a right
angle. And remember area grows by k^2, not by
k.
A toy car is a real car shrunk by a scale factor. If the model is built at "1 to 18", every
length on the real car is 18 times longer than on the model — the
wheels, the doors, the bonnet, all multiplied by the same number, so the model looks exactly
like the real thing in miniature. But careful with paint: the real car's surface needs
18^2 = 324 times as much paint, because area scales by the square
of the scale factor.
→ scale factor 18 →