Similar triangles — area ratio = (similarity ratio)²

Dependencies: Similar triangles: cevian ratios = scale (corresponding altitude ratio = $k$), Triangle area = ½ × base × height (area = $\tfrac{1}{2}\cdot$ base $\cdot$ height).

Statement

Let $\triangle ABC \sim \triangle A'B'C'$ with similarity ratio $k$ (i.e. $AB / A'B' = BC / B'C' = CA / C'A' = k$). Denote the areas of the two triangles by $S$ and $S'$ respectively. Then

That is, the area ratio of similar triangles equals the square of the similarity ratio.