Have you ever enlarged or shrunk a photo? The original photo and the enlarged one differ in size, but the shape stays exactly the same. In math, this relationship is called "similarity." On the other hand, a shape that matches perfectly when laid on top of another — same size and same shape — is called "congruent."
Telling similarity and congruence apart can be confusing at first. If every pair of corresponding angles between two shapes is equal, you've already satisfied one of the conditions for similarity, and if the ratio of side lengths matches too, they're fully similar. If that ratio happens to be 1:1 — meaning even the size matches — that's congruence. So you could say congruence is a special case of similarity.
What makes this concept interesting is how area changes as you adjust the scale factor. Double the side lengths and the area doesn't double — it becomes 4 times (2²) as large; triple them and the area becomes 9 times (3²) as large. For a solid shape with volume, tripling the sides makes the volume a whopping 27 times (3³) larger. Calculate a map or a scale drawing without knowing this "magic of squares and cubes," and you can end up with a serious error.
In everyday life, similarity is essential in mapmaking, architectural blueprints, and building model cars or plastic model kits. To figure out a real building's actual size from a blueprint scaled down at a ratio of 1:100, you need the concept of a scale factor. When a photo lab enlarges a print and the proportions don't match, the photo ends up looking oddly stretched — that's similarity breaking down, too.
In middle school, you also learn the three conditions for determining whether two triangles are similar: when the ratio of all three sides is equal, when the ratio of two sides is equal and the angle between them matches, and when two angles each match. The condition "similar if two angles match" is especially useful because a triangle's three interior angles always add up to 180 degrees — so if just two angles match, the third has to match automatically too. That means when solving an actual problem, you only need to check two angles, which makes the calculation much simpler. The famous method of measuring a tree's or a building's height using its shadow uses exactly this principle. Knowing just a stick's height and the length of its shadow, you can measure the shadow a large building casts at that same moment and use the scale factor to calculate the building's actual height.
When working with kids, I'd recommend drawing the same shape at several different scale factors and having them count for themselves how many times larger the area gets. At first they'll guess "I scaled it 2×, so the area should be 2× too" — and when they actually count and find it's 4×, that surprise tends to stick in memory for a long time. On our activity page, you can move the scale slider and watch how area and perimeter change in real time. I think letting kids experience firsthand the moment their first guess turns out different from the actual result is the surest way — better than any explanation — to make the squared-and-cubed rule of similarity stick. Drawing a scaled-down floor plan of your own backyard or classroom floor together, and going back and forth calculating the real length versus the length on the drawing, is also a great activity for getting a feel for the scale-factor concept in your bones.