Similarity – Same Shape, Different Size

Similar shapes have the same shape but not necessarily the same size. They are like scaled copies of each other. Understanding similarity is key in map reading, scale models, photography, and many areas of engineering.

One of the oldest recorded uses of similar triangles comes from the Greek philosopher Thales of Miletus, who is said to have measured the height of the Great Pyramid of Giza around 600 BCE simply by comparing the length of its shadow to the shadow cast by a vertical stick of known height at the same moment — the two shadow-and-object pairs form similar triangles, so the ratio of shadow to height is the same for both. The same principle of similarity is why a road map printed at a scale of 1:100,000 is still useful even though it is nowhere near the size of the real landscape, and why architects and engineers build small-scale physical models of buildings and bridges before construction begins, confident that the proportions — and therefore the way forces and stresses behave — scale predictably between the model and the real structure.

What Does Similar Mean?

Proportion Through Similar Shapes

Every side of the large rectangle is exactly double the matching side of the small one (8 = 2 × 4, and 6 = 2 × 3), so 4:3 and 8:6 are equivalent ratios — the rectangles are proportional, and mathematicians call them similar. This is exactly the idea behind scale drawings and proportional relationships: enlarging or shrinking a shape while keeping every ratio of sides identical.

Congruence vs Similarity

Scale Factor

The scale factor k = (length on image) / (length on original). If k > 1, the shape is enlarged. If k < 1, it is reduced. If k = 1, the shapes are congruent.

Similar Triangles – Conditions

Area and Volume Scale Factors

Seeing why area scales by k²: enlarge a square by a linear scale factor of k = 2 (doubling each side), and the new square isn't just twice as big — it's four times as big. You can literally see why: the enlarged square is exactly big enough to fit 4 copies of the original square inside it.

The big square holds exactly 4 small squares — and 4 = 2². That's the area scale factor. If you scaled a cube the same way (doubling every edge), it would take 8 = 2³ copies of the small cube to fill the large one, which is exactly why volume scales by k³.

Key Takeaways

Practice: Scale Factors